src/HOL/Typedef.thy
 author paulson Mon Feb 22 14:37:56 2016 +0000 (2016-02-22) changeset 62379 340738057c8c parent 61799 4cf66f21b764 child 63434 c956d995bec6 permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
1 (*  Title:      HOL/Typedef.thy
2     Author:     Markus Wenzel, TU Munich
3 *)
5 section \<open>HOL type definitions\<close>
7 theory Typedef
8 imports Set
9 keywords "typedef" :: thy_goal and "morphisms"
10 begin
12 locale type_definition =
13   fixes Rep and Abs and A
14   assumes Rep: "Rep x \<in> A"
15     and Rep_inverse: "Abs (Rep x) = x"
16     and Abs_inverse: "y \<in> A \<Longrightarrow> Rep (Abs y) = y"
17   \<comment> \<open>This will be axiomatized for each typedef!\<close>
18 begin
20 lemma Rep_inject: "Rep x = Rep y \<longleftrightarrow> x = y"
21 proof
22   assume "Rep x = Rep y"
23   then have "Abs (Rep x) = Abs (Rep y)" by (simp only:)
24   moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
25   moreover have "Abs (Rep y) = y" by (rule Rep_inverse)
26   ultimately show "x = y" by simp
27 next
28   assume "x = y"
29   then show "Rep x = Rep y" by (simp only:)
30 qed
32 lemma Abs_inject:
33   assumes "x \<in> A" and "y \<in> A"
34   shows "Abs x = Abs y \<longleftrightarrow> x = y"
35 proof
36   assume "Abs x = Abs y"
37   then have "Rep (Abs x) = Rep (Abs y)" by (simp only:)
38   moreover from \<open>x \<in> A\<close> have "Rep (Abs x) = x" by (rule Abs_inverse)
39   moreover from \<open>y \<in> A\<close> have "Rep (Abs y) = y" by (rule Abs_inverse)
40   ultimately show "x = y" by simp
41 next
42   assume "x = y"
43   then show "Abs x = Abs y" by (simp only:)
44 qed
46 lemma Rep_cases [cases set]:
47   assumes "y \<in> A"
48     and hyp: "\<And>x. y = Rep x \<Longrightarrow> P"
49   shows P
50 proof (rule hyp)
51   from \<open>y \<in> A\<close> have "Rep (Abs y) = y" by (rule Abs_inverse)
52   then show "y = Rep (Abs y)" ..
53 qed
55 lemma Abs_cases [cases type]:
56   assumes r: "\<And>y. x = Abs y \<Longrightarrow> y \<in> A \<Longrightarrow> P"
57   shows P
58 proof (rule r)
59   have "Abs (Rep x) = x" by (rule Rep_inverse)
60   then show "x = Abs (Rep x)" ..
61   show "Rep x \<in> A" by (rule Rep)
62 qed
64 lemma Rep_induct [induct set]:
65   assumes y: "y \<in> A"
66     and hyp: "\<And>x. P (Rep x)"
67   shows "P y"
68 proof -
69   have "P (Rep (Abs y))" by (rule hyp)
70   moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)
71   ultimately show "P y" by simp
72 qed
74 lemma Abs_induct [induct type]:
75   assumes r: "\<And>y. y \<in> A \<Longrightarrow> P (Abs y)"
76   shows "P x"
77 proof -
78   have "Rep x \<in> A" by (rule Rep)
79   then have "P (Abs (Rep x))" by (rule r)
80   moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
81   ultimately show "P x" by simp
82 qed
84 lemma Rep_range: "range Rep = A"
85 proof
86   show "range Rep \<subseteq> A" using Rep by (auto simp add: image_def)
87   show "A \<subseteq> range Rep"
88   proof
89     fix x assume "x \<in> A"
90     then have "x = Rep (Abs x)" by (rule Abs_inverse [symmetric])
91     then show "x \<in> range Rep" by (rule range_eqI)
92   qed
93 qed
95 lemma Abs_image: "Abs ` A = UNIV"
96 proof
97   show "Abs ` A \<subseteq> UNIV" by (rule subset_UNIV)
98   show "UNIV \<subseteq> Abs ` A"
99   proof
100     fix x
101     have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
102     moreover have "Rep x \<in> A" by (rule Rep)
103     ultimately show "x \<in> Abs ` A" by (rule image_eqI)
104   qed
105 qed
107 end
109 ML_file "Tools/typedef.ML"
111 end