src/HOL/Typedef.thy
 author paulson Mon May 23 15:33:24 2016 +0100 (2016-05-23) changeset 63114 27afe7af7379 parent 61799 4cf66f21b764 child 63434 c956d995bec6 permissions -rw-r--r--
Lots of new material for multivariate analysis
```     1 (*  Title:      HOL/Typedef.thy
```
```     2     Author:     Markus Wenzel, TU Munich
```
```     3 *)
```
```     4
```
```     5 section \<open>HOL type definitions\<close>
```
```     6
```
```     7 theory Typedef
```
```     8 imports Set
```
```     9 keywords "typedef" :: thy_goal and "morphisms"
```
```    10 begin
```
```    11
```
```    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
```
```    19
```
```    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
```
```    31
```
```    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
```
```    45
```
```    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
```
```    54
```
```    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
```
```    63
```
```    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
```
```    73
```
```    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
```
```    83
```
```    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
```
```    94
```
```    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
```
```   106
```
```   107 end
```
```   108
```
```   109 ML_file "Tools/typedef.ML"
```
```   110
```
```   111 end
```