src/HOL/Typedef.thy
 author nipkow Mon Jan 30 21:49:41 2012 +0100 (2012-01-30) changeset 46372 6fa9cdb8b850 parent 41732 996b0c14a430 child 46947 b8c7eb0c2f89 permissions -rw-r--r--
 wenzelm@11608 ` 1` ```(* Title: HOL/Typedef.thy ``` wenzelm@11608 ` 2` ``` Author: Markus Wenzel, TU Munich ``` wenzelm@11743 ` 3` ```*) ``` wenzelm@11608 ` 4` wenzelm@11979 ` 5` ```header {* HOL type definitions *} ``` wenzelm@11608 ` 6` nipkow@15131 ` 7` ```theory Typedef ``` nipkow@15140 ` 8` ```imports Set ``` haftmann@38536 ` 9` ```uses ("Tools/typedef.ML") ``` nipkow@15131 ` 10` ```begin ``` wenzelm@11608 ` 11` wenzelm@13412 ` 12` ```locale type_definition = ``` wenzelm@13412 ` 13` ``` fixes Rep and Abs and A ``` wenzelm@13412 ` 14` ``` assumes Rep: "Rep x \ A" ``` wenzelm@13412 ` 15` ``` and Rep_inverse: "Abs (Rep x) = x" ``` wenzelm@13412 ` 16` ``` and Abs_inverse: "y \ A ==> Rep (Abs y) = y" ``` wenzelm@13412 ` 17` ``` -- {* This will be axiomatized for each typedef! *} ``` haftmann@23247 ` 18` ```begin ``` wenzelm@11608 ` 19` haftmann@23247 ` 20` ```lemma Rep_inject: ``` wenzelm@13412 ` 21` ``` "(Rep x = Rep y) = (x = y)" ``` wenzelm@13412 ` 22` ```proof ``` wenzelm@13412 ` 23` ``` assume "Rep x = Rep y" ``` haftmann@23710 ` 24` ``` then have "Abs (Rep x) = Abs (Rep y)" by (simp only:) ``` haftmann@23710 ` 25` ``` moreover have "Abs (Rep x) = x" by (rule Rep_inverse) ``` haftmann@23710 ` 26` ``` moreover have "Abs (Rep y) = y" by (rule Rep_inverse) ``` haftmann@23710 ` 27` ``` ultimately show "x = y" by simp ``` wenzelm@13412 ` 28` ```next ``` wenzelm@13412 ` 29` ``` assume "x = y" ``` wenzelm@13412 ` 30` ``` thus "Rep x = Rep y" by (simp only:) ``` wenzelm@13412 ` 31` ```qed ``` wenzelm@11608 ` 32` haftmann@23247 ` 33` ```lemma Abs_inject: ``` wenzelm@13412 ` 34` ``` assumes x: "x \ A" and y: "y \ A" ``` wenzelm@13412 ` 35` ``` shows "(Abs x = Abs y) = (x = y)" ``` wenzelm@13412 ` 36` ```proof ``` wenzelm@13412 ` 37` ``` assume "Abs x = Abs y" ``` haftmann@23710 ` 38` ``` then have "Rep (Abs x) = Rep (Abs y)" by (simp only:) ``` haftmann@23710 ` 39` ``` moreover from x have "Rep (Abs x) = x" by (rule Abs_inverse) ``` haftmann@23710 ` 40` ``` moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse) ``` haftmann@23710 ` 41` ``` ultimately show "x = y" by simp ``` wenzelm@13412 ` 42` ```next ``` wenzelm@13412 ` 43` ``` assume "x = y" ``` wenzelm@13412 ` 44` ``` thus "Abs x = Abs y" by (simp only:) ``` wenzelm@11608 ` 45` ```qed ``` wenzelm@11608 ` 46` haftmann@23247 ` 47` ```lemma Rep_cases [cases set]: ``` wenzelm@13412 ` 48` ``` assumes y: "y \ A" ``` wenzelm@13412 ` 49` ``` and hyp: "!!x. y = Rep x ==> P" ``` wenzelm@13412 ` 50` ``` shows P ``` wenzelm@13412 ` 51` ```proof (rule hyp) ``` wenzelm@13412 ` 52` ``` from y have "Rep (Abs y) = y" by (rule Abs_inverse) ``` wenzelm@13412 ` 53` ``` thus "y = Rep (Abs y)" .. ``` wenzelm@11608 ` 54` ```qed ``` wenzelm@11608 ` 55` haftmann@23247 ` 56` ```lemma Abs_cases [cases type]: ``` wenzelm@13412 ` 57` ``` assumes r: "!!y. x = Abs y ==> y \ A ==> P" ``` wenzelm@13412 ` 58` ``` shows P ``` wenzelm@13412 ` 59` ```proof (rule r) ``` wenzelm@13412 ` 60` ``` have "Abs (Rep x) = x" by (rule Rep_inverse) ``` wenzelm@13412 ` 61` ``` thus "x = Abs (Rep x)" .. ``` wenzelm@13412 ` 62` ``` show "Rep x \ A" by (rule Rep) ``` wenzelm@11608 ` 63` ```qed ``` wenzelm@11608 ` 64` haftmann@23247 ` 65` ```lemma Rep_induct [induct set]: ``` wenzelm@13412 ` 66` ``` assumes y: "y \ A" ``` wenzelm@13412 ` 67` ``` and hyp: "!!x. P (Rep x)" ``` wenzelm@13412 ` 68` ``` shows "P y" ``` wenzelm@11608 ` 69` ```proof - ``` wenzelm@13412 ` 70` ``` have "P (Rep (Abs y))" by (rule hyp) ``` haftmann@23710 ` 71` ``` moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse) ``` haftmann@23710 ` 72` ``` ultimately show "P y" by simp ``` wenzelm@11608 ` 73` ```qed ``` wenzelm@11608 ` 74` haftmann@23247 ` 75` ```lemma Abs_induct [induct type]: ``` wenzelm@13412 ` 76` ``` assumes r: "!!y. y \ A ==> P (Abs y)" ``` wenzelm@13412 ` 77` ``` shows "P x" ``` wenzelm@11608 ` 78` ```proof - ``` wenzelm@13412 ` 79` ``` have "Rep x \ A" by (rule Rep) ``` haftmann@23710 ` 80` ``` then have "P (Abs (Rep x))" by (rule r) ``` haftmann@23710 ` 81` ``` moreover have "Abs (Rep x) = x" by (rule Rep_inverse) ``` haftmann@23710 ` 82` ``` ultimately show "P x" by simp ``` wenzelm@11608 ` 83` ```qed ``` wenzelm@11608 ` 84` huffman@27295 ` 85` ```lemma Rep_range: "range Rep = A" ``` huffman@24269 ` 86` ```proof ``` huffman@24269 ` 87` ``` show "range Rep <= A" using Rep by (auto simp add: image_def) ``` huffman@24269 ` 88` ``` show "A <= range Rep" ``` nipkow@23433 ` 89` ``` proof ``` nipkow@23433 ` 90` ``` fix x assume "x : A" ``` huffman@24269 ` 91` ``` hence "x = Rep (Abs x)" by (rule Abs_inverse [symmetric]) ``` huffman@24269 ` 92` ``` thus "x : range Rep" by (rule range_eqI) ``` nipkow@23433 ` 93` ``` qed ``` nipkow@23433 ` 94` ```qed ``` nipkow@23433 ` 95` huffman@27295 ` 96` ```lemma Abs_image: "Abs ` A = UNIV" ``` huffman@27295 ` 97` ```proof ``` huffman@27295 ` 98` ``` show "Abs ` A <= UNIV" by (rule subset_UNIV) ``` huffman@27295 ` 99` ```next ``` huffman@27295 ` 100` ``` show "UNIV <= Abs ` A" ``` huffman@27295 ` 101` ``` proof ``` huffman@27295 ` 102` ``` fix x ``` huffman@27295 ` 103` ``` have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric]) ``` huffman@27295 ` 104` ``` moreover have "Rep x : A" by (rule Rep) ``` huffman@27295 ` 105` ``` ultimately show "x : Abs ` A" by (rule image_eqI) ``` huffman@27295 ` 106` ``` qed ``` huffman@27295 ` 107` ```qed ``` huffman@27295 ` 108` haftmann@23247 ` 109` ```end ``` haftmann@23247 ` 110` haftmann@31723 ` 111` ```use "Tools/typedef.ML" setup Typedef.setup ``` wenzelm@11608 ` 112` wenzelm@11608 ` 113` ```end ```