src/HOL/Typedef.thy
author webertj
Fri, 19 Oct 2012 15:12:52 +0200
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(*  Title:      HOL/Typedef.thy
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    Author:     Markus Wenzel, TU Munich
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*)
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header {* HOL type definitions *}
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theory Typedef
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imports Set
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keywords "typedef" :: thy_goal and "morphisms"
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begin
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locale type_definition =
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  fixes Rep and Abs and A
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  assumes Rep: "Rep x \<in> A"
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    and Rep_inverse: "Abs (Rep x) = x"
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    and Abs_inverse: "y \<in> A ==> Rep (Abs y) = y"
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  -- {* This will be axiomatized for each typedef! *}
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begin
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lemma Rep_inject:
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  "(Rep x = Rep y) = (x = y)"
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proof
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  assume "Rep x = Rep y"
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  then have "Abs (Rep x) = Abs (Rep y)" by (simp only:)
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  moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
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  moreover have "Abs (Rep y) = y" by (rule Rep_inverse)
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  ultimately show "x = y" by simp
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next
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  assume "x = y"
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  thus "Rep x = Rep y" by (simp only:)
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qed
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lemma Abs_inject:
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  assumes x: "x \<in> A" and y: "y \<in> A"
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  shows "(Abs x = Abs y) = (x = y)"
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proof
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  assume "Abs x = Abs y"
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  then have "Rep (Abs x) = Rep (Abs y)" by (simp only:)
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  moreover from x have "Rep (Abs x) = x" by (rule Abs_inverse)
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  moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)
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  ultimately show "x = y" by simp
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next
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  assume "x = y"
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  thus "Abs x = Abs y" by (simp only:)
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qed
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lemma Rep_cases [cases set]:
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  assumes y: "y \<in> A"
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    and hyp: "!!x. y = Rep x ==> P"
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  shows P
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proof (rule hyp)
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  from y have "Rep (Abs y) = y" by (rule Abs_inverse)
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  thus "y = Rep (Abs y)" ..
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qed
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lemma Abs_cases [cases type]:
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  assumes r: "!!y. x = Abs y ==> y \<in> A ==> P"
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  shows P
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proof (rule r)
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  have "Abs (Rep x) = x" by (rule Rep_inverse)
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  thus "x = Abs (Rep x)" ..
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  show "Rep x \<in> A" by (rule Rep)
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qed
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lemma Rep_induct [induct set]:
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  assumes y: "y \<in> A"
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    and hyp: "!!x. P (Rep x)"
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  shows "P y"
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proof -
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  have "P (Rep (Abs y))" by (rule hyp)
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  moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)
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  ultimately show "P y" by simp
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qed
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lemma Abs_induct [induct type]:
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  assumes r: "!!y. y \<in> A ==> P (Abs y)"
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  shows "P x"
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proof -
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  have "Rep x \<in> A" by (rule Rep)
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  then have "P (Abs (Rep x))" by (rule r)
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  moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
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  ultimately show "P x" by simp
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qed
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lemma Rep_range: "range Rep = A"
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proof
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  show "range Rep <= A" using Rep by (auto simp add: image_def)
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  show "A <= range Rep"
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  proof
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    fix x assume "x : A"
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    hence "x = Rep (Abs x)" by (rule Abs_inverse [symmetric])
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    thus "x : range Rep" by (rule range_eqI)
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  qed
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qed
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lemma Abs_image: "Abs ` A = UNIV"
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proof
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  show "Abs ` A <= UNIV" by (rule subset_UNIV)
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next
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  show "UNIV <= Abs ` A"
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  proof
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    fix x
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    have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
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    moreover have "Rep x : A" by (rule Rep)
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    ultimately show "x : Abs ` A" by (rule image_eqI)
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  qed
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qed
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end
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ML_file "Tools/typedef.ML" setup Typedef.setup
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end