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src/HOL/Typedef.thy

author | wenzelm |

Thu Sep 27 22:26:00 2001 +0200 (2001-09-27) | |

changeset 11608 | c760ea8154ee |

child 11654 | 53d18ab990f6 |

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

renamed theory "subset" to "Typedef";

1 (* Title: HOL/Typedef.thy

2 ID: $Id$

3 Author: Markus Wenzel, TU Munich

5 Misc set-theory lemmas and HOL type definitions.

6 *)

8 theory Typedef = Set

9 files "subset.ML" "equalities.ML" "mono.ML"

10 "Tools/induct_attrib.ML" ("Tools/typedef_package.ML"):

12 (** belongs to theory Ord **)

14 theorems linorder_cases [case_names less equal greater] =

15 linorder_less_split

17 (* Courtesy of Stephan Merz *)

18 lemma Least_mono:

19 "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y

20 ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"

21 apply clarify

22 apply (erule_tac P = "%x. x : S" in LeastI2)

23 apply fast

24 apply (rule LeastI2)

25 apply (auto elim: monoD intro!: order_antisym)

26 done

29 (*belongs to theory Set*)

30 setup Rulify.setup

33 section {* HOL type definitions *}

35 constdefs

36 type_definition :: "('a => 'b) => ('b => 'a) => 'b set => bool"

37 "type_definition Rep Abs A ==

38 (\<forall>x. Rep x \<in> A) \<and>

39 (\<forall>x. Abs (Rep x) = x) \<and>

40 (\<forall>y \<in> A. Rep (Abs y) = y)"

41 -- {* This will be stated as an axiom for each typedef! *}

43 lemma type_definitionI [intro]:

44 "(!!x. Rep x \<in> A) ==>

45 (!!x. Abs (Rep x) = x) ==>

46 (!!y. y \<in> A ==> Rep (Abs y) = y) ==>

47 type_definition Rep Abs A"

48 by (unfold type_definition_def) blast

50 theorem Rep: "type_definition Rep Abs A ==> Rep x \<in> A"

51 by (unfold type_definition_def) blast

53 theorem Rep_inverse: "type_definition Rep Abs A ==> Abs (Rep x) = x"

54 by (unfold type_definition_def) blast

56 theorem Abs_inverse: "type_definition Rep Abs A ==> y \<in> A ==> Rep (Abs y) = y"

57 by (unfold type_definition_def) blast

59 theorem Rep_inject: "type_definition Rep Abs A ==> (Rep x = Rep y) = (x = y)"

60 proof -

61 assume tydef: "type_definition Rep Abs A"

62 show ?thesis

63 proof

64 assume "Rep x = Rep y"

65 hence "Abs (Rep x) = Abs (Rep y)" by (simp only:)

66 thus "x = y" by (simp only: Rep_inverse [OF tydef])

67 next

68 assume "x = y"

69 thus "Rep x = Rep y" by simp

70 qed

71 qed

73 theorem Abs_inject:

74 "type_definition Rep Abs A ==> x \<in> A ==> y \<in> A ==> (Abs x = Abs y) = (x = y)"

75 proof -

76 assume tydef: "type_definition Rep Abs A"

77 assume x: "x \<in> A" and y: "y \<in> A"

78 show ?thesis

79 proof

80 assume "Abs x = Abs y"

81 hence "Rep (Abs x) = Rep (Abs y)" by simp

82 moreover from x have "Rep (Abs x) = x" by (rule Abs_inverse [OF tydef])

83 moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])

84 ultimately show "x = y" by (simp only:)

85 next

86 assume "x = y"

87 thus "Abs x = Abs y" by simp

88 qed

89 qed

91 theorem Rep_cases:

92 "type_definition Rep Abs A ==> y \<in> A ==> (!!x. y = Rep x ==> P) ==> P"

93 proof -

94 assume tydef: "type_definition Rep Abs A"

95 assume y: "y \<in> A" and r: "(!!x. y = Rep x ==> P)"

96 show P

97 proof (rule r)

98 from y have "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])

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

100 qed

101 qed

103 theorem Abs_cases:

104 "type_definition Rep Abs A ==> (!!y. x = Abs y ==> y \<in> A ==> P) ==> P"

105 proof -

106 assume tydef: "type_definition Rep Abs A"

107 assume r: "!!y. x = Abs y ==> y \<in> A ==> P"

108 show P

109 proof (rule r)

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

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

112 show "Rep x \<in> A" by (rule Rep [OF tydef])

113 qed

114 qed

116 theorem Rep_induct:

117 "type_definition Rep Abs A ==> y \<in> A ==> (!!x. P (Rep x)) ==> P y"

118 proof -

119 assume tydef: "type_definition Rep Abs A"

120 assume "!!x. P (Rep x)" hence "P (Rep (Abs y))" .

121 moreover assume "y \<in> A" hence "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])

122 ultimately show "P y" by (simp only:)

123 qed

125 theorem Abs_induct:

126 "type_definition Rep Abs A ==> (!!y. y \<in> A ==> P (Abs y)) ==> P x"

127 proof -

128 assume tydef: "type_definition Rep Abs A"

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

130 have "Rep x \<in> A" by (rule Rep [OF tydef])

131 hence "P (Abs (Rep x))" by (rule r)

132 moreover have "Abs (Rep x) = x" by (rule Rep_inverse [OF tydef])

133 ultimately show "P x" by (simp only:)

134 qed

136 setup InductAttrib.setup

137 use "Tools/typedef_package.ML"

139 end