src/HOL/Typedef.thy
 author haftmann Thu Aug 12 17:56:41 2010 +0200 (2010-08-12) changeset 38393 7c045c03598f parent 37863 7f113caabcf4 child 38536 7e57a0dcbd4f permissions -rw-r--r--
group record-related ML files
 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@20426 ` 9` ```uses ``` haftmann@31723 ` 10` ``` ("Tools/typedef.ML") ``` haftmann@20426 ` 11` ``` ("Tools/typedef_codegen.ML") ``` nipkow@15131 ` 12` ```begin ``` wenzelm@11608 ` 13` haftmann@23247 ` 14` ```ML {* ``` wenzelm@37863 ` 15` ```structure HOL = struct val thy = @{theory HOL} end; ``` haftmann@23247 ` 16` ```*} -- "belongs to theory HOL" ``` haftmann@23247 ` 17` wenzelm@13412 ` 18` ```locale type_definition = ``` wenzelm@13412 ` 19` ``` fixes Rep and Abs and A ``` wenzelm@13412 ` 20` ``` assumes Rep: "Rep x \ A" ``` wenzelm@13412 ` 21` ``` and Rep_inverse: "Abs (Rep x) = x" ``` wenzelm@13412 ` 22` ``` and Abs_inverse: "y \ A ==> Rep (Abs y) = y" ``` wenzelm@13412 ` 23` ``` -- {* This will be axiomatized for each typedef! *} ``` haftmann@23247 ` 24` ```begin ``` wenzelm@11608 ` 25` haftmann@23247 ` 26` ```lemma Rep_inject: ``` wenzelm@13412 ` 27` ``` "(Rep x = Rep y) = (x = y)" ``` wenzelm@13412 ` 28` ```proof ``` wenzelm@13412 ` 29` ``` assume "Rep x = Rep y" ``` haftmann@23710 ` 30` ``` then have "Abs (Rep x) = Abs (Rep y)" by (simp only:) ``` haftmann@23710 ` 31` ``` moreover have "Abs (Rep x) = x" by (rule Rep_inverse) ``` haftmann@23710 ` 32` ``` moreover have "Abs (Rep y) = y" by (rule Rep_inverse) ``` haftmann@23710 ` 33` ``` ultimately show "x = y" by simp ``` wenzelm@13412 ` 34` ```next ``` wenzelm@13412 ` 35` ``` assume "x = y" ``` wenzelm@13412 ` 36` ``` thus "Rep x = Rep y" by (simp only:) ``` wenzelm@13412 ` 37` ```qed ``` wenzelm@11608 ` 38` haftmann@23247 ` 39` ```lemma Abs_inject: ``` wenzelm@13412 ` 40` ``` assumes x: "x \ A" and y: "y \ A" ``` wenzelm@13412 ` 41` ``` shows "(Abs x = Abs y) = (x = y)" ``` wenzelm@13412 ` 42` ```proof ``` wenzelm@13412 ` 43` ``` assume "Abs x = Abs y" ``` haftmann@23710 ` 44` ``` then have "Rep (Abs x) = Rep (Abs y)" by (simp only:) ``` haftmann@23710 ` 45` ``` moreover from x have "Rep (Abs x) = x" by (rule Abs_inverse) ``` haftmann@23710 ` 46` ``` moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse) ``` haftmann@23710 ` 47` ``` ultimately show "x = y" by simp ``` wenzelm@13412 ` 48` ```next ``` wenzelm@13412 ` 49` ``` assume "x = y" ``` wenzelm@13412 ` 50` ``` thus "Abs x = Abs y" by (simp only:) ``` wenzelm@11608 ` 51` ```qed ``` wenzelm@11608 ` 52` haftmann@23247 ` 53` ```lemma Rep_cases [cases set]: ``` wenzelm@13412 ` 54` ``` assumes y: "y \ A" ``` wenzelm@13412 ` 55` ``` and hyp: "!!x. y = Rep x ==> P" ``` wenzelm@13412 ` 56` ``` shows P ``` wenzelm@13412 ` 57` ```proof (rule hyp) ``` wenzelm@13412 ` 58` ``` from y have "Rep (Abs y) = y" by (rule Abs_inverse) ``` wenzelm@13412 ` 59` ``` thus "y = Rep (Abs y)" .. ``` wenzelm@11608 ` 60` ```qed ``` wenzelm@11608 ` 61` haftmann@23247 ` 62` ```lemma Abs_cases [cases type]: ``` wenzelm@13412 ` 63` ``` assumes r: "!!y. x = Abs y ==> y \ A ==> P" ``` wenzelm@13412 ` 64` ``` shows P ``` wenzelm@13412 ` 65` ```proof (rule r) ``` wenzelm@13412 ` 66` ``` have "Abs (Rep x) = x" by (rule Rep_inverse) ``` wenzelm@13412 ` 67` ``` thus "x = Abs (Rep x)" .. ``` wenzelm@13412 ` 68` ``` show "Rep x \ A" by (rule Rep) ``` wenzelm@11608 ` 69` ```qed ``` wenzelm@11608 ` 70` haftmann@23247 ` 71` ```lemma Rep_induct [induct set]: ``` wenzelm@13412 ` 72` ``` assumes y: "y \ A" ``` wenzelm@13412 ` 73` ``` and hyp: "!!x. P (Rep x)" ``` wenzelm@13412 ` 74` ``` shows "P y" ``` wenzelm@11608 ` 75` ```proof - ``` wenzelm@13412 ` 76` ``` have "P (Rep (Abs y))" by (rule hyp) ``` haftmann@23710 ` 77` ``` moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse) ``` haftmann@23710 ` 78` ``` ultimately show "P y" by simp ``` wenzelm@11608 ` 79` ```qed ``` wenzelm@11608 ` 80` haftmann@23247 ` 81` ```lemma Abs_induct [induct type]: ``` wenzelm@13412 ` 82` ``` assumes r: "!!y. y \ A ==> P (Abs y)" ``` wenzelm@13412 ` 83` ``` shows "P x" ``` wenzelm@11608 ` 84` ```proof - ``` wenzelm@13412 ` 85` ``` have "Rep x \ A" by (rule Rep) ``` haftmann@23710 ` 86` ``` then have "P (Abs (Rep x))" by (rule r) ``` haftmann@23710 ` 87` ``` moreover have "Abs (Rep x) = x" by (rule Rep_inverse) ``` haftmann@23710 ` 88` ``` ultimately show "P x" by simp ``` wenzelm@11608 ` 89` ```qed ``` wenzelm@11608 ` 90` huffman@27295 ` 91` ```lemma Rep_range: "range Rep = A" ``` huffman@24269 ` 92` ```proof ``` huffman@24269 ` 93` ``` show "range Rep <= A" using Rep by (auto simp add: image_def) ``` huffman@24269 ` 94` ``` show "A <= range Rep" ``` nipkow@23433 ` 95` ``` proof ``` nipkow@23433 ` 96` ``` fix x assume "x : A" ``` huffman@24269 ` 97` ``` hence "x = Rep (Abs x)" by (rule Abs_inverse [symmetric]) ``` huffman@24269 ` 98` ``` thus "x : range Rep" by (rule range_eqI) ``` nipkow@23433 ` 99` ``` qed ``` nipkow@23433 ` 100` ```qed ``` nipkow@23433 ` 101` huffman@27295 ` 102` ```lemma Abs_image: "Abs ` A = UNIV" ``` huffman@27295 ` 103` ```proof ``` huffman@27295 ` 104` ``` show "Abs ` A <= UNIV" by (rule subset_UNIV) ``` huffman@27295 ` 105` ```next ``` huffman@27295 ` 106` ``` show "UNIV <= Abs ` A" ``` huffman@27295 ` 107` ``` proof ``` huffman@27295 ` 108` ``` fix x ``` huffman@27295 ` 109` ``` have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric]) ``` huffman@27295 ` 110` ``` moreover have "Rep x : A" by (rule Rep) ``` huffman@27295 ` 111` ``` ultimately show "x : Abs ` A" by (rule image_eqI) ``` huffman@27295 ` 112` ``` qed ``` huffman@27295 ` 113` ```qed ``` huffman@27295 ` 114` haftmann@23247 ` 115` ```end ``` haftmann@23247 ` 116` haftmann@31723 ` 117` ```use "Tools/typedef.ML" setup Typedef.setup ``` wenzelm@29056 ` 118` ```use "Tools/typedef_codegen.ML" setup TypedefCodegen.setup ``` wenzelm@11608 ` 119` wenzelm@11608 ` 120` ```end ```