src/HOL/Typedef.thy
author wenzelm
Mon Dec 07 10:38:04 2015 +0100 (2015-12-07)
changeset 61799 4cf66f21b764
parent 61102 0ec9fd8d8119
child 63434 c956d995bec6
permissions -rw-r--r--
isabelle update_cartouches -c -t;
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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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section \<open>HOL type definitions\<close>
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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 \<Longrightarrow> Rep (Abs y) = y"
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  \<comment> \<open>This will be axiomatized for each typedef!\<close>
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begin
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lemma Rep_inject: "Rep x = Rep y \<longleftrightarrow> 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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  then show "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 \<in> A" and "y \<in> A"
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  shows "Abs x = Abs y \<longleftrightarrow> 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 \<open>x \<in> A\<close> have "Rep (Abs x) = x" by (rule Abs_inverse)
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  moreover from \<open>y \<in> A\<close> 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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  then show "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 \<in> A"
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    and hyp: "\<And>x. y = Rep x \<Longrightarrow> P"
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  shows P
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proof (rule hyp)
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  from \<open>y \<in> A\<close> have "Rep (Abs y) = y" by (rule Abs_inverse)
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  then show "y = Rep (Abs y)" ..
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qed
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lemma Abs_cases [cases type]:
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  assumes r: "\<And>y. x = Abs y \<Longrightarrow> y \<in> A \<Longrightarrow> 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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  then show "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: "\<And>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: "\<And>y. y \<in> A \<Longrightarrow> 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 \<subseteq> A" using Rep by (auto simp add: image_def)
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  show "A \<subseteq> range Rep"
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  proof
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    fix x assume "x \<in> A"
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    then have "x = Rep (Abs x)" by (rule Abs_inverse [symmetric])
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    then show "x \<in> 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 \<subseteq> UNIV" by (rule subset_UNIV)
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  show "UNIV \<subseteq> 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 \<in> A" by (rule Rep)
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    ultimately show "x \<in> 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"
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end