11608

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(* Title: HOL/Typedef.thy


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ID: $Id$


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Author: Markus Wenzel, TU Munich


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Misc settheory lemmas and HOL type definitions.


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*)


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theory Typedef = Set


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files "subset.ML" "equalities.ML" "mono.ML"


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"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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(* Courtesy of Stephan Merz *)


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lemma Least_mono:


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"mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y


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==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"


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apply clarify


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apply (erule_tac P = "%x. x : S" in LeastI2)


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apply fast


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apply (rule LeastI2)


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apply (auto elim: monoD intro!: order_antisym)


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done


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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 from x have "Rep (Abs x) = x" by (rule Abs_inverse [OF tydef])


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moreover from y have "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
