src/HOL/subset.thy
author nipkow
Fri, 24 Nov 2000 16:49:27 +0100
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(*  Title:      HOL/subset.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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Subset lemmas and HOL type definitions.
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*)
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theory subset = Set
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files "Tools/induct_attrib.ML" ("Tools/typedef_package.ML"):
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(*belongs to theory Ord*)
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theorems linorder_cases [case_names less equal greater] =
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  linorder_less_split
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(*belongs to theory Set*)
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setup Rulify.setup
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section {* HOL type definitions *}
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constdefs
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  type_definition :: "('a => 'b) => ('b => 'a) => 'b set => bool"
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  "type_definition Rep Abs A ==
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    (\<forall>x. Rep x \<in> A) \<and>
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    (\<forall>x. Abs (Rep x) = x) \<and>
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    (\<forall>y \<in> A. Rep (Abs y) = y)"
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  -- {* This will be stated as an axiom for each typedef! *}
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lemma type_definitionI [intro]:
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  "(!!x. Rep x \<in> A) ==>
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    (!!x. Abs (Rep x) = x) ==>
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    (!!y. y \<in> A ==> Rep (Abs y) = y) ==>
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    type_definition Rep Abs A"
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  by (unfold type_definition_def) blast
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theorem Rep: "type_definition Rep Abs A ==> Rep x \<in> A"
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  by (unfold type_definition_def) blast
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theorem Rep_inverse: "type_definition Rep Abs A ==> Abs (Rep x) = x"
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  by (unfold type_definition_def) blast
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theorem Abs_inverse: "type_definition Rep Abs A ==> y \<in> A ==> Rep (Abs y) = y"
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  by (unfold type_definition_def) blast
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theorem Rep_inject: "type_definition Rep Abs A ==> (Rep x = Rep y) = (x = y)"
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proof -
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  assume tydef: "type_definition Rep Abs A"
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  show ?thesis
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  proof
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    assume "Rep x = Rep y"
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    hence "Abs (Rep x) = Abs (Rep y)" by (simp only:)
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    thus "x = y" by (simp only: Rep_inverse [OF tydef])
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  next
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    assume "x = y"
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    thus "Rep x = Rep y" by simp
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  qed
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qed
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theorem Abs_inject:
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  "type_definition Rep Abs A ==> x \<in> A ==> y \<in> A ==> (Abs x = Abs y) = (x = y)"
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proof -
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  assume tydef: "type_definition Rep Abs A"
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  assume x: "x \<in> A" and y: "y \<in> A"
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  show ?thesis
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  proof
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    assume "Abs x = Abs y"
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    hence "Rep (Abs x) = Rep (Abs y)" by simp
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    moreover note x hence "Rep (Abs x) = x" by (rule Abs_inverse [OF tydef])
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    moreover note y hence "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])
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    ultimately show "x = y" by (simp only:)
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  next
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    assume "x = y"
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    thus "Abs x = Abs y" by simp
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  qed
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qed
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theorem Rep_cases:
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  "type_definition Rep Abs A ==> y \<in> A ==> (!!x. y = Rep x ==> P) ==> P"
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proof -
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  assume tydef: "type_definition Rep Abs A"
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  assume y: "y \<in> A" and r: "(!!x. y = Rep x ==> P)"
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  show P
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  proof (rule r)
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    from y have "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])
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    thus "y = Rep (Abs y)" ..
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  qed
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qed
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theorem Abs_cases:
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  "type_definition Rep Abs A ==> (!!y. x = Abs y ==> y \<in> A ==> P) ==> P"
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proof -
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  assume tydef: "type_definition Rep Abs A"
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  assume r: "!!y. x = Abs y ==> y \<in> A ==> P"
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  show P
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  proof (rule r)
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    have "Abs (Rep x) = x" by (rule Rep_inverse [OF tydef])
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    thus "x = Abs (Rep x)" ..
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    show "Rep x \<in> A" by (rule Rep [OF tydef])
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  qed
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qed
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theorem Rep_induct:
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  "type_definition Rep Abs A ==> y \<in> A ==> (!!x. P (Rep x)) ==> P y"
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proof -
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  assume tydef: "type_definition Rep Abs A"
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  assume "!!x. P (Rep x)" hence "P (Rep (Abs y))" .
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  moreover assume "y \<in> A" hence "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])
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  ultimately show "P y" by (simp only:)
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qed
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theorem Abs_induct:
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  "type_definition Rep Abs A ==> (!!y. y \<in> A ==> P (Abs y)) ==> P x"
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proof -
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  assume tydef: "type_definition Rep Abs A"
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  assume r: "!!y. y \<in> A ==> P (Abs y)"
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  have "Rep x \<in> A" by (rule Rep [OF tydef])
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  hence "P (Abs (Rep x))" by (rule r)
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  moreover have "Abs (Rep x) = x" by (rule Rep_inverse [OF tydef])
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  ultimately show "P x" by (simp only:)
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qed
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setup InductAttrib.setup
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use "Tools/typedef_package.ML"
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end