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src/HOL/Typedef.thy

changeset 13412 | 666137b488a4 |

parent 12023 | d982f98e0f0d |

child 13421 | 8fcdf4a26468 |

1.1 --- a/src/HOL/Typedef.thy Wed Jul 24 00:09:44 2002 +0200 1.2 +++ b/src/HOL/Typedef.thy Wed Jul 24 00:10:52 2002 +0200 1.3 @@ -8,105 +8,79 @@ 1.4 theory Typedef = Set 1.5 files ("Tools/typedef_package.ML"): 1.6 1.7 -constdefs 1.8 - type_definition :: "('a => 'b) => ('b => 'a) => 'b set => bool" 1.9 - "type_definition Rep Abs A == 1.10 - (\<forall>x. Rep x \<in> A) \<and> 1.11 - (\<forall>x. Abs (Rep x) = x) \<and> 1.12 - (\<forall>y \<in> A. Rep (Abs y) = y)" 1.13 - -- {* This will be stated as an axiom for each typedef! *} 1.14 +locale type_definition = 1.15 + fixes Rep and Abs and A 1.16 + assumes Rep: "Rep x \<in> A" 1.17 + and Rep_inverse: "Abs (Rep x) = x" 1.18 + and Abs_inverse: "y \<in> A ==> Rep (Abs y) = y" 1.19 + -- {* This will be axiomatized for each typedef! *} 1.20 1.21 -lemma type_definitionI [intro]: 1.22 - "(!!x. Rep x \<in> A) ==> 1.23 - (!!x. Abs (Rep x) = x) ==> 1.24 - (!!y. y \<in> A ==> Rep (Abs y) = y) ==> 1.25 - type_definition Rep Abs A" 1.26 - by (unfold type_definition_def) blast 1.27 - 1.28 -theorem Rep: "type_definition Rep Abs A ==> Rep x \<in> A" 1.29 - by (unfold type_definition_def) blast 1.30 +lemmas type_definitionI [intro] = 1.31 + type_definition.intro [OF type_definition_axioms.intro] 1.32 1.33 -theorem Rep_inverse: "type_definition Rep Abs A ==> Abs (Rep x) = x" 1.34 - by (unfold type_definition_def) blast 1.35 - 1.36 -theorem Abs_inverse: "type_definition Rep Abs A ==> y \<in> A ==> Rep (Abs y) = y" 1.37 - by (unfold type_definition_def) blast 1.38 +lemma (in type_definition) Rep_inject: 1.39 + "(Rep x = Rep y) = (x = y)" 1.40 +proof 1.41 + assume "Rep x = Rep y" 1.42 + hence "Abs (Rep x) = Abs (Rep y)" by (simp only:) 1.43 + also have "Abs (Rep x) = x" by (rule Rep_inverse) 1.44 + also have "Abs (Rep y) = y" by (rule Rep_inverse) 1.45 + finally show "x = y" . 1.46 +next 1.47 + assume "x = y" 1.48 + thus "Rep x = Rep y" by (simp only:) 1.49 +qed 1.50 1.51 -theorem Rep_inject: "type_definition Rep Abs A ==> (Rep x = Rep y) = (x = y)" 1.52 -proof - 1.53 - assume tydef: "type_definition Rep Abs A" 1.54 - show ?thesis 1.55 - proof 1.56 - assume "Rep x = Rep y" 1.57 - hence "Abs (Rep x) = Abs (Rep y)" by (simp only:) 1.58 - thus "x = y" by (simp only: Rep_inverse [OF tydef]) 1.59 - next 1.60 - assume "x = y" 1.61 - thus "Rep x = Rep y" by simp 1.62 - qed 1.63 +lemma (in type_definition) Abs_inject: 1.64 + assumes x: "x \<in> A" and y: "y \<in> A" 1.65 + shows "(Abs x = Abs y) = (x = y)" 1.66 +proof 1.67 + assume "Abs x = Abs y" 1.68 + hence "Rep (Abs x) = Rep (Abs y)" by (simp only:) 1.69 + also from x have "Rep (Abs x) = x" by (rule Abs_inverse) 1.70 + also from y have "Rep (Abs y) = y" by (rule Abs_inverse) 1.71 + finally show "x = y" . 1.72 +next 1.73 + assume "x = y" 1.74 + thus "Abs x = Abs y" by (simp only:) 1.75 qed 1.76 1.77 -theorem Abs_inject: 1.78 - "type_definition Rep Abs A ==> x \<in> A ==> y \<in> A ==> (Abs x = Abs y) = (x = y)" 1.79 -proof - 1.80 - assume tydef: "type_definition Rep Abs A" 1.81 - assume x: "x \<in> A" and y: "y \<in> A" 1.82 - show ?thesis 1.83 - proof 1.84 - assume "Abs x = Abs y" 1.85 - hence "Rep (Abs x) = Rep (Abs y)" by simp 1.86 - moreover from x have "Rep (Abs x) = x" by (rule Abs_inverse [OF tydef]) 1.87 - moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef]) 1.88 - ultimately show "x = y" by (simp only:) 1.89 - next 1.90 - assume "x = y" 1.91 - thus "Abs x = Abs y" by simp 1.92 - qed 1.93 +lemma (in type_definition) Rep_cases [cases set]: 1.94 + assumes y: "y \<in> A" 1.95 + and hyp: "!!x. y = Rep x ==> P" 1.96 + shows P 1.97 +proof (rule hyp) 1.98 + from y have "Rep (Abs y) = y" by (rule Abs_inverse) 1.99 + thus "y = Rep (Abs y)" .. 1.100 qed 1.101 1.102 -theorem Rep_cases: 1.103 - "type_definition Rep Abs A ==> y \<in> A ==> (!!x. y = Rep x ==> P) ==> P" 1.104 -proof - 1.105 - assume tydef: "type_definition Rep Abs A" 1.106 - assume y: "y \<in> A" and r: "(!!x. y = Rep x ==> P)" 1.107 - show P 1.108 - proof (rule r) 1.109 - from y have "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef]) 1.110 - thus "y = Rep (Abs y)" .. 1.111 - qed 1.112 +lemma (in type_definition) Abs_cases [cases type]: 1.113 + assumes r: "!!y. x = Abs y ==> y \<in> A ==> P" 1.114 + shows P 1.115 +proof (rule r) 1.116 + have "Abs (Rep x) = x" by (rule Rep_inverse) 1.117 + thus "x = Abs (Rep x)" .. 1.118 + show "Rep x \<in> A" by (rule Rep) 1.119 qed 1.120 1.121 -theorem Abs_cases: 1.122 - "type_definition Rep Abs A ==> (!!y. x = Abs y ==> y \<in> A ==> P) ==> P" 1.123 +lemma (in type_definition) Rep_induct [induct set]: 1.124 + assumes y: "y \<in> A" 1.125 + and hyp: "!!x. P (Rep x)" 1.126 + shows "P y" 1.127 proof - 1.128 - assume tydef: "type_definition Rep Abs A" 1.129 - assume r: "!!y. x = Abs y ==> y \<in> A ==> P" 1.130 - show P 1.131 - proof (rule r) 1.132 - have "Abs (Rep x) = x" by (rule Rep_inverse [OF tydef]) 1.133 - thus "x = Abs (Rep x)" .. 1.134 - show "Rep x \<in> A" by (rule Rep [OF tydef]) 1.135 - qed 1.136 + have "P (Rep (Abs y))" by (rule hyp) 1.137 + also from y have "Rep (Abs y) = y" by (rule Abs_inverse) 1.138 + finally show "P y" . 1.139 qed 1.140 1.141 -theorem Rep_induct: 1.142 - "type_definition Rep Abs A ==> y \<in> A ==> (!!x. P (Rep x)) ==> P y" 1.143 +lemma (in type_definition) Abs_induct [induct type]: 1.144 + assumes r: "!!y. y \<in> A ==> P (Abs y)" 1.145 + shows "P x" 1.146 proof - 1.147 - assume tydef: "type_definition Rep Abs A" 1.148 - assume "!!x. P (Rep x)" hence "P (Rep (Abs y))" . 1.149 - moreover assume "y \<in> A" hence "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef]) 1.150 - ultimately show "P y" by (simp only:) 1.151 -qed 1.152 - 1.153 -theorem Abs_induct: 1.154 - "type_definition Rep Abs A ==> (!!y. y \<in> A ==> P (Abs y)) ==> P x" 1.155 -proof - 1.156 - assume tydef: "type_definition Rep Abs A" 1.157 - assume r: "!!y. y \<in> A ==> P (Abs y)" 1.158 - have "Rep x \<in> A" by (rule Rep [OF tydef]) 1.159 + have "Rep x \<in> A" by (rule Rep) 1.160 hence "P (Abs (Rep x))" by (rule r) 1.161 - moreover have "Abs (Rep x) = x" by (rule Rep_inverse [OF tydef]) 1.162 - ultimately show "P x" by (simp only:) 1.163 + also have "Abs (Rep x) = x" by (rule Rep_inverse) 1.164 + finally show "P x" . 1.165 qed 1.166 1.167 use "Tools/typedef_package.ML"