src/HOL/Typedef.thy
 author blanchet Tue Nov 07 15:16:42 2017 +0100 (19 months ago) changeset 67022 49309fe530fd parent 63434 c956d995bec6 child 69605 a96320074298 permissions -rw-r--r--
more robust parsing for THF proofs (esp. polymorphic Leo-III proofs)
```     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
```
```    10   "typedef" :: thy_goal and
```
```    11   "morphisms" :: quasi_command
```
```    12 begin
```
```    13
```
```    14 locale type_definition =
```
```    15   fixes Rep and Abs and A
```
```    16   assumes Rep: "Rep x \<in> A"
```
```    17     and Rep_inverse: "Abs (Rep x) = x"
```
```    18     and Abs_inverse: "y \<in> A \<Longrightarrow> Rep (Abs y) = y"
```
```    19   \<comment> \<open>This will be axiomatized for each typedef!\<close>
```
```    20 begin
```
```    21
```
```    22 lemma Rep_inject: "Rep x = Rep y \<longleftrightarrow> x = y"
```
```    23 proof
```
```    24   assume "Rep x = Rep y"
```
```    25   then have "Abs (Rep x) = Abs (Rep y)" by (simp only:)
```
```    26   moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    27   moreover have "Abs (Rep y) = y" by (rule Rep_inverse)
```
```    28   ultimately show "x = y" by simp
```
```    29 next
```
```    30   assume "x = y"
```
```    31   then show "Rep x = Rep y" by (simp only:)
```
```    32 qed
```
```    33
```
```    34 lemma Abs_inject:
```
```    35   assumes "x \<in> A" and "y \<in> A"
```
```    36   shows "Abs x = Abs y \<longleftrightarrow> x = y"
```
```    37 proof
```
```    38   assume "Abs x = Abs y"
```
```    39   then have "Rep (Abs x) = Rep (Abs y)" by (simp only:)
```
```    40   moreover from \<open>x \<in> A\<close> have "Rep (Abs x) = x" by (rule Abs_inverse)
```
```    41   moreover from \<open>y \<in> A\<close> have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    42   ultimately show "x = y" by simp
```
```    43 next
```
```    44   assume "x = y"
```
```    45   then show "Abs x = Abs y" by (simp only:)
```
```    46 qed
```
```    47
```
```    48 lemma Rep_cases [cases set]:
```
```    49   assumes "y \<in> A"
```
```    50     and hyp: "\<And>x. y = Rep x \<Longrightarrow> P"
```
```    51   shows P
```
```    52 proof (rule hyp)
```
```    53   from \<open>y \<in> A\<close> have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    54   then show "y = Rep (Abs y)" ..
```
```    55 qed
```
```    56
```
```    57 lemma Abs_cases [cases type]:
```
```    58   assumes r: "\<And>y. x = Abs y \<Longrightarrow> y \<in> A \<Longrightarrow> P"
```
```    59   shows P
```
```    60 proof (rule r)
```
```    61   have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    62   then show "x = Abs (Rep x)" ..
```
```    63   show "Rep x \<in> A" by (rule Rep)
```
```    64 qed
```
```    65
```
```    66 lemma Rep_induct [induct set]:
```
```    67   assumes y: "y \<in> A"
```
```    68     and hyp: "\<And>x. P (Rep x)"
```
```    69   shows "P y"
```
```    70 proof -
```
```    71   have "P (Rep (Abs y))" by (rule hyp)
```
```    72   moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    73   ultimately show "P y" by simp
```
```    74 qed
```
```    75
```
```    76 lemma Abs_induct [induct type]:
```
```    77   assumes r: "\<And>y. y \<in> A \<Longrightarrow> P (Abs y)"
```
```    78   shows "P x"
```
```    79 proof -
```
```    80   have "Rep x \<in> A" by (rule Rep)
```
```    81   then have "P (Abs (Rep x))" by (rule r)
```
```    82   moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    83   ultimately show "P x" by simp
```
```    84 qed
```
```    85
```
```    86 lemma Rep_range: "range Rep = A"
```
```    87 proof
```
```    88   show "range Rep \<subseteq> A" using Rep by (auto simp add: image_def)
```
```    89   show "A \<subseteq> range Rep"
```
```    90   proof
```
```    91     fix x assume "x \<in> A"
```
```    92     then have "x = Rep (Abs x)" by (rule Abs_inverse [symmetric])
```
```    93     then show "x \<in> range Rep" by (rule range_eqI)
```
```    94   qed
```
```    95 qed
```
```    96
```
```    97 lemma Abs_image: "Abs ` A = UNIV"
```
```    98 proof
```
```    99   show "Abs ` A \<subseteq> UNIV" by (rule subset_UNIV)
```
```   100   show "UNIV \<subseteq> Abs ` A"
```
```   101   proof
```
```   102     fix x
```
```   103     have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
```
```   104     moreover have "Rep x \<in> A" by (rule Rep)
```
```   105     ultimately show "x \<in> Abs ` A" by (rule image_eqI)
```
```   106   qed
```
```   107 qed
```
```   108
```
```   109 end
```
```   110
```
```   111 ML_file "Tools/typedef.ML"
```
```   112
```
```   113 end
```