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theory Typedef(* Title: HOL/Typedef.thy ID: $Id: Typedef.thy,v 1.14 2005/06/17 14:13:05 haftmann Exp $ Author: Markus Wenzel, TU Munich *) header {* HOL type definitions *} theory Typedef imports Set uses ("Tools/typedef_package.ML") begin locale type_definition = fixes Rep and Abs and A assumes Rep: "Rep x ∈ A" and Rep_inverse: "Abs (Rep x) = x" and Abs_inverse: "y ∈ A ==> Rep (Abs y) = y" -- {* This will be axiomatized for each typedef! *} lemma (in type_definition) Rep_inject: "(Rep x = Rep y) = (x = y)" proof assume "Rep x = Rep y" hence "Abs (Rep x) = Abs (Rep y)" by (simp only:) also have "Abs (Rep x) = x" by (rule Rep_inverse) also have "Abs (Rep y) = y" by (rule Rep_inverse) finally show "x = y" . next assume "x = y" thus "Rep x = Rep y" by (simp only:) qed lemma (in type_definition) Abs_inject: assumes x: "x ∈ A" and y: "y ∈ A" shows "(Abs x = Abs y) = (x = y)" proof assume "Abs x = Abs y" hence "Rep (Abs x) = Rep (Abs y)" by (simp only:) also from x have "Rep (Abs x) = x" by (rule Abs_inverse) also from y have "Rep (Abs y) = y" by (rule Abs_inverse) finally show "x = y" . next assume "x = y" thus "Abs x = Abs y" by (simp only:) qed lemma (in type_definition) Rep_cases [cases set]: assumes y: "y ∈ 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 (in type_definition) Abs_cases [cases type]: assumes r: "!!y. x = Abs y ==> y ∈ A ==> P" shows P proof (rule r) have "Abs (Rep x) = x" by (rule Rep_inverse) thus "x = Abs (Rep x)" .. show "Rep x ∈ A" by (rule Rep) qed lemma (in type_definition) Rep_induct [induct set]: assumes y: "y ∈ A" and hyp: "!!x. P (Rep x)" shows "P y" proof - have "P (Rep (Abs y))" by (rule hyp) also from y have "Rep (Abs y) = y" by (rule Abs_inverse) finally show "P y" . qed lemma (in type_definition) Abs_induct [induct type]: assumes r: "!!y. y ∈ A ==> P (Abs y)" shows "P x" proof - have "Rep x ∈ A" by (rule Rep) hence "P (Abs (Rep x))" by (rule r) also have "Abs (Rep x) = x" by (rule Rep_inverse) finally show "P x" . qed use "Tools/typedef_package.ML" setup TypedefPackage.setup end
lemma Rep_inject:
type_definition Rep Abs A ==> (Rep x = Rep y) = (x = y)
lemma Abs_inject:
[| type_definition Rep Abs A; x ∈ A; y ∈ A |] ==> (Abs x = Abs y) = (x = y)
lemma Rep_cases:
[| type_definition Rep Abs A; y ∈ A; !!x. y = Rep x ==> P |] ==> P
lemma Abs_cases:
[| type_definition Rep Abs A; !!y. [| x = Abs y; y ∈ A |] ==> P |] ==> P
lemma Rep_induct:
[| type_definition Rep Abs A; y ∈ A; !!x. P (Rep x) |] ==> P y
lemma Abs_induct:
[| type_definition Rep Abs A; !!y. y ∈ A ==> P (Abs y) |] ==> P x