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src/HOL/Typedef.thy

author | haftmann |

Mon Nov 29 13:44:54 2010 +0100 (2010-11-29) | |

changeset 40815 | 6e2d17cc0d1d |

parent 38536 | 7e57a0dcbd4f |

child 41732 | 996b0c14a430 |

permissions | -rw-r--r-- |

equivI has replaced equiv.intro

1 (* Title: HOL/Typedef.thy

2 Author: Markus Wenzel, TU Munich

3 *)

5 header {* HOL type definitions *}

7 theory Typedef

8 imports Set

9 uses ("Tools/typedef.ML")

10 begin

12 ML {*

13 structure HOL = struct val thy = @{theory HOL} end;

14 *} -- "belongs to theory HOL"

16 locale type_definition =

17 fixes Rep and Abs and A

18 assumes Rep: "Rep x \<in> A"

19 and Rep_inverse: "Abs (Rep x) = x"

20 and Abs_inverse: "y \<in> A ==> Rep (Abs y) = y"

21 -- {* This will be axiomatized for each typedef! *}

22 begin

24 lemma Rep_inject:

25 "(Rep x = Rep y) = (x = y)"

26 proof

27 assume "Rep x = Rep y"

28 then have "Abs (Rep x) = Abs (Rep y)" by (simp only:)

29 moreover have "Abs (Rep x) = x" by (rule Rep_inverse)

30 moreover have "Abs (Rep y) = y" by (rule Rep_inverse)

31 ultimately show "x = y" by simp

32 next

33 assume "x = y"

34 thus "Rep x = Rep y" by (simp only:)

35 qed

37 lemma Abs_inject:

38 assumes x: "x \<in> A" and y: "y \<in> A"

39 shows "(Abs x = Abs y) = (x = y)"

40 proof

41 assume "Abs x = Abs y"

42 then have "Rep (Abs x) = Rep (Abs y)" by (simp only:)

43 moreover from x have "Rep (Abs x) = x" by (rule Abs_inverse)

44 moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)

45 ultimately show "x = y" by simp

46 next

47 assume "x = y"

48 thus "Abs x = Abs y" by (simp only:)

49 qed

51 lemma Rep_cases [cases set]:

52 assumes y: "y \<in> A"

53 and hyp: "!!x. y = Rep x ==> P"

54 shows P

55 proof (rule hyp)

56 from y have "Rep (Abs y) = y" by (rule Abs_inverse)

57 thus "y = Rep (Abs y)" ..

58 qed

60 lemma Abs_cases [cases type]:

61 assumes r: "!!y. x = Abs y ==> y \<in> A ==> P"

62 shows P

63 proof (rule r)

64 have "Abs (Rep x) = x" by (rule Rep_inverse)

65 thus "x = Abs (Rep x)" ..

66 show "Rep x \<in> A" by (rule Rep)

67 qed

69 lemma Rep_induct [induct set]:

70 assumes y: "y \<in> A"

71 and hyp: "!!x. P (Rep x)"

72 shows "P y"

73 proof -

74 have "P (Rep (Abs y))" by (rule hyp)

75 moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)

76 ultimately show "P y" by simp

77 qed

79 lemma Abs_induct [induct type]:

80 assumes r: "!!y. y \<in> A ==> P (Abs y)"

81 shows "P x"

82 proof -

83 have "Rep x \<in> A" by (rule Rep)

84 then have "P (Abs (Rep x))" by (rule r)

85 moreover have "Abs (Rep x) = x" by (rule Rep_inverse)

86 ultimately show "P x" by simp

87 qed

89 lemma Rep_range: "range Rep = A"

90 proof

91 show "range Rep <= A" using Rep by (auto simp add: image_def)

92 show "A <= range Rep"

93 proof

94 fix x assume "x : A"

95 hence "x = Rep (Abs x)" by (rule Abs_inverse [symmetric])

96 thus "x : range Rep" by (rule range_eqI)

97 qed

98 qed

100 lemma Abs_image: "Abs ` A = UNIV"

101 proof

102 show "Abs ` A <= UNIV" by (rule subset_UNIV)

103 next

104 show "UNIV <= Abs ` A"

105 proof

106 fix x

107 have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])

108 moreover have "Rep x : A" by (rule Rep)

109 ultimately show "x : Abs ` A" by (rule image_eqI)

110 qed

111 qed

113 end

115 use "Tools/typedef.ML" setup Typedef.setup

117 end