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src/HOL/Typedef.thy

author | wenzelm |

Sun, 02 Nov 2014 18:21:45 +0100 | |

changeset 58889 | 5b7a9633cfa8 |

parent 58239 | 1c5bc387bd4c |

child 60758 | d8d85a8172b5 |

permissions | -rw-r--r-- |

modernized header uniformly as section;

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