1 (* Title: HOL/Fun |
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2 ID: $Id$ |
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3 Author: Tobias Nipkow, Cambridge University Computer Laboratory |
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4 Copyright 1993 University of Cambridge |
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5 |
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6 Lemmas about functions. |
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7 *) |
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8 |
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9 goal Fun.thy "(f = g) = (!x. f(x)=g(x))"; |
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10 by (rtac iffI 1); |
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11 by(asm_simp_tac HOL_ss 1); |
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12 by(rtac ext 1 THEN asm_simp_tac HOL_ss 1); |
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13 qed "expand_fun_eq"; |
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14 |
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15 val prems = goal Fun.thy |
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16 "[| f(x)=u; !!x. P(x) ==> g(f(x)) = x; P(x) |] ==> x=g(u)"; |
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17 by (rtac (arg_cong RS box_equals) 1); |
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18 by (REPEAT (resolve_tac (prems@[refl]) 1)); |
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19 qed "apply_inverse"; |
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20 |
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21 |
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22 (*** Range of a function ***) |
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23 |
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24 (*Frequently b does not have the syntactic form of f(x).*) |
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25 val [prem] = goalw Fun.thy [range_def] "b=f(x) ==> b : range(f)"; |
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26 by (EVERY1 [rtac CollectI, rtac exI, rtac prem]); |
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27 qed "range_eqI"; |
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28 |
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29 val rangeI = refl RS range_eqI; |
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30 |
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31 val [major,minor] = goalw Fun.thy [range_def] |
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32 "[| b : range(%x.f(x)); !!x. b=f(x) ==> P |] ==> P"; |
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33 by (rtac (major RS CollectD RS exE) 1); |
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34 by (etac minor 1); |
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35 qed "rangeE"; |
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36 |
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37 (*** Image of a set under a function ***) |
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38 |
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39 val prems = goalw Fun.thy [image_def] "[| b=f(x); x:A |] ==> b : f``A"; |
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40 by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1)); |
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41 qed "image_eqI"; |
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42 |
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43 val imageI = refl RS image_eqI; |
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44 |
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45 (*The eta-expansion gives variable-name preservation.*) |
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46 val major::prems = goalw Fun.thy [image_def] |
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47 "[| b : (%x.f(x))``A; !!x.[| b=f(x); x:A |] ==> P |] ==> P"; |
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48 by (rtac (major RS CollectD RS bexE) 1); |
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49 by (REPEAT (ares_tac prems 1)); |
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50 qed "imageE"; |
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51 |
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52 goalw Fun.thy [o_def] "(f o g)``r = f``(g``r)"; |
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53 by (rtac set_ext 1); |
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54 by (fast_tac (HOL_cs addIs [imageI] addSEs [imageE]) 1); |
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55 qed "image_compose"; |
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56 |
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57 goal Fun.thy "f``(A Un B) = f``A Un f``B"; |
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58 by (rtac set_ext 1); |
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59 by (fast_tac (HOL_cs addIs [imageI,UnCI] addSEs [imageE,UnE]) 1); |
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60 qed "image_Un"; |
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61 |
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62 (*** inj(f): f is a one-to-one function ***) |
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63 |
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64 val prems = goalw Fun.thy [inj_def] |
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65 "[| !! x y. f(x) = f(y) ==> x=y |] ==> inj(f)"; |
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66 by (fast_tac (HOL_cs addIs prems) 1); |
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67 qed "injI"; |
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68 |
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69 val [major] = goal Fun.thy "(!!x. g(f(x)) = x) ==> inj(f)"; |
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70 by (rtac injI 1); |
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71 by (etac (arg_cong RS box_equals) 1); |
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72 by (rtac major 1); |
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73 by (rtac major 1); |
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74 qed "inj_inverseI"; |
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75 |
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76 val [major,minor] = goalw Fun.thy [inj_def] |
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77 "[| inj(f); f(x) = f(y) |] ==> x=y"; |
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78 by (rtac (major RS spec RS spec RS mp) 1); |
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79 by (rtac minor 1); |
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80 qed "injD"; |
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81 |
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82 (*Useful with the simplifier*) |
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83 val [major] = goal Fun.thy "inj(f) ==> (f(x) = f(y)) = (x=y)"; |
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84 by (rtac iffI 1); |
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85 by (etac (major RS injD) 1); |
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86 by (etac arg_cong 1); |
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87 qed "inj_eq"; |
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88 |
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89 val [major] = goal Fun.thy "inj(f) ==> (@x.f(x)=f(y)) = y"; |
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90 by (rtac (major RS injD) 1); |
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91 by (rtac selectI 1); |
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92 by (rtac refl 1); |
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93 qed "inj_select"; |
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94 |
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95 (*A one-to-one function has an inverse (given using select).*) |
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96 val [major] = goalw Fun.thy [Inv_def] "inj(f) ==> Inv(f,f(x)) = x"; |
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97 by (EVERY1 [rtac (major RS inj_select)]); |
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98 qed "Inv_f_f"; |
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99 |
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100 (* Useful??? *) |
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101 val [oneone,minor] = goal Fun.thy |
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102 "[| inj(f); !!y. y: range(f) ==> P(Inv(f,y)) |] ==> P(x)"; |
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103 by (res_inst_tac [("t", "x")] (oneone RS (Inv_f_f RS subst)) 1); |
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104 by (rtac (rangeI RS minor) 1); |
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105 qed "inj_transfer"; |
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106 |
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107 |
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108 (*** inj_onto(f,A): f is one-to-one over A ***) |
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109 |
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110 val prems = goalw Fun.thy [inj_onto_def] |
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111 "(!! x y. [| f(x) = f(y); x:A; y:A |] ==> x=y) ==> inj_onto(f,A)"; |
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112 by (fast_tac (HOL_cs addIs prems addSIs [ballI]) 1); |
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113 qed "inj_ontoI"; |
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114 |
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115 val [major] = goal Fun.thy |
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116 "(!!x. x:A ==> g(f(x)) = x) ==> inj_onto(f,A)"; |
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117 by (rtac inj_ontoI 1); |
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118 by (etac (apply_inverse RS trans) 1); |
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119 by (REPEAT (eresolve_tac [asm_rl,major] 1)); |
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120 qed "inj_onto_inverseI"; |
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121 |
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122 val major::prems = goalw Fun.thy [inj_onto_def] |
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123 "[| inj_onto(f,A); f(x)=f(y); x:A; y:A |] ==> x=y"; |
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124 by (rtac (major RS bspec RS bspec RS mp) 1); |
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125 by (REPEAT (resolve_tac prems 1)); |
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126 qed "inj_ontoD"; |
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127 |
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128 goal Fun.thy "!!x y.[| inj_onto(f,A); x:A; y:A |] ==> (f(x)=f(y)) = (x=y)"; |
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129 by (fast_tac (HOL_cs addSEs [inj_ontoD]) 1); |
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130 qed "inj_onto_iff"; |
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131 |
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132 val major::prems = goal Fun.thy |
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133 "[| inj_onto(f,A); ~x=y; x:A; y:A |] ==> ~ f(x)=f(y)"; |
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134 by (rtac contrapos 1); |
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135 by (etac (major RS inj_ontoD) 2); |
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136 by (REPEAT (resolve_tac prems 1)); |
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137 qed "inj_onto_contraD"; |
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138 |
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139 |
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140 (*** Lemmas about inj ***) |
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141 |
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142 val prems = goalw Fun.thy [o_def] |
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143 "[| inj(f); inj_onto(g,range(f)) |] ==> inj(g o f)"; |
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144 by (cut_facts_tac prems 1); |
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145 by (fast_tac (HOL_cs addIs [injI,rangeI] |
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146 addEs [injD,inj_ontoD]) 1); |
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147 qed "comp_inj"; |
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148 |
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149 val [prem] = goal Fun.thy "inj(f) ==> inj_onto(f,A)"; |
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150 by (fast_tac (HOL_cs addIs [prem RS injD, inj_ontoI]) 1); |
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151 qed "inj_imp"; |
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152 |
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153 val [prem] = goalw Fun.thy [Inv_def] "y : range(f) ==> f(Inv(f,y)) = y"; |
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154 by (EVERY1 [rtac (prem RS rangeE), rtac selectI, etac sym]); |
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155 qed "f_Inv_f"; |
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156 |
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157 val prems = goal Fun.thy |
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158 "[| Inv(f,x)=Inv(f,y); x: range(f); y: range(f) |] ==> x=y"; |
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159 by (rtac (arg_cong RS box_equals) 1); |
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160 by (REPEAT (resolve_tac (prems @ [f_Inv_f]) 1)); |
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161 qed "Inv_injective"; |
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162 |
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163 val prems = goal Fun.thy |
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164 "[| inj(f); A<=range(f) |] ==> inj_onto(Inv(f), A)"; |
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165 by (cut_facts_tac prems 1); |
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166 by (fast_tac (HOL_cs addIs [inj_ontoI] |
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167 addEs [Inv_injective,injD,subsetD]) 1); |
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168 qed "inj_onto_Inv"; |
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169 |
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170 |
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171 (*** Set reasoning tools ***) |
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172 |
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173 val set_cs = HOL_cs |
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174 addSIs [ballI, PowI, subsetI, InterI, INT_I, INT1_I, CollectI, |
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175 ComplI, IntI, DiffI, UnCI, insertCI] |
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176 addIs [bexI, UnionI, UN_I, UN1_I, imageI, rangeI] |
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177 addSEs [bexE, make_elim PowD, UnionE, UN_E, UN1_E, DiffE, |
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178 CollectE, ComplE, IntE, UnE, insertE, imageE, rangeE, emptyE] |
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179 addEs [ballE, InterD, InterE, INT_D, INT_E, make_elim INT1_D, |
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180 subsetD, subsetCE]; |
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181 |
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182 fun cfast_tac prems = cut_facts_tac prems THEN' fast_tac set_cs; |
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183 |
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184 |
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185 fun prover s = prove_goal Fun.thy s (fn _=>[fast_tac set_cs 1]); |
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186 |
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187 val mem_simps = map prover |
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188 [ "(a : A Un B) = (a:A | a:B)", |
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189 "(a : A Int B) = (a:A & a:B)", |
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190 "(a : Compl(B)) = (~a:B)", |
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191 "(a : A-B) = (a:A & ~a:B)", |
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192 "(a : {b}) = (a=b)", |
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193 "(a : {x.P(x)}) = P(a)" ]; |
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194 |
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195 val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs; |
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196 |
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197 val set_ss = |
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198 HOL_ss addsimps mem_simps |
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199 addcongs [ball_cong,bex_cong] |
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200 setmksimps (mksimps mksimps_pairs); |
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