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src/HOL/Typedef.thy

author | nipkow |

Fri Mar 06 17:38:47 2009 +0100 (2009-03-06) | |

changeset 30313 | b2441b0c8d38 |

parent 29797 | 08ef36ed2f8a |

child 31723 | f5cafe803b55 |

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

added lemmas

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

10 ("Tools/typedef_package.ML")

11 ("Tools/typecopy_package.ML")

12 ("Tools/typedef_codegen.ML")

13 begin

15 ML {*

16 structure HOL = struct val thy = theory "HOL" end;

17 *} -- "belongs to theory HOL"

19 locale type_definition =

20 fixes Rep and Abs and A

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

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

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

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

25 begin

27 lemma Rep_inject:

28 "(Rep x = Rep y) = (x = y)"

29 proof

30 assume "Rep x = Rep y"

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

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

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

34 ultimately show "x = y" by simp

35 next

36 assume "x = y"

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

38 qed

40 lemma Abs_inject:

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

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

43 proof

44 assume "Abs x = Abs y"

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

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

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

48 ultimately show "x = y" by simp

49 next

50 assume "x = y"

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

52 qed

54 lemma Rep_cases [cases set]:

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

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

57 shows P

58 proof (rule hyp)

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

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

61 qed

63 lemma Abs_cases [cases type]:

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

65 shows P

66 proof (rule r)

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

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

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

70 qed

72 lemma Rep_induct [induct set]:

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

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

75 shows "P y"

76 proof -

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

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

79 ultimately show "P y" by simp

80 qed

82 lemma Abs_induct [induct type]:

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

84 shows "P x"

85 proof -

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

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

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

89 ultimately show "P x" by simp

90 qed

92 lemma Rep_range: "range Rep = A"

93 proof

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

95 show "A <= range Rep"

96 proof

97 fix x assume "x : A"

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

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

100 qed

101 qed

103 lemma Abs_image: "Abs ` A = UNIV"

104 proof

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

106 next

107 show "UNIV <= Abs ` A"

108 proof

109 fix x

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

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

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

113 qed

114 qed

116 end

118 use "Tools/typedef_package.ML" setup TypedefPackage.setup

119 use "Tools/typecopy_package.ML" setup TypecopyPackage.setup

120 use "Tools/typedef_codegen.ML" setup TypedefCodegen.setup

122 end