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1 (* Title: HOLCF/sprod2.ML |
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2 ID: $Id$ |
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3 Author: Franz Regensburger |
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4 Copyright 1993 Technische Universitaet Muenchen |
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5 |
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6 Lemmas for sprod2.thy |
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7 *) |
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8 |
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9 |
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10 open Sprod2; |
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11 |
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12 (* ------------------------------------------------------------------------ *) |
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13 (* access to less_sprod in class po *) |
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14 (* ------------------------------------------------------------------------ *) |
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15 |
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16 val less_sprod3a = prove_goal Sprod2.thy |
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17 "p1=Ispair(UU,UU) ==> p1 << p2" |
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18 (fn prems => |
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19 [ |
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20 (cut_facts_tac prems 1), |
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21 (rtac (inst_sprod_po RS ssubst) 1), |
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22 (etac less_sprod1a 1) |
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23 ]); |
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24 |
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25 |
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26 val less_sprod3b = prove_goal Sprod2.thy |
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27 "~p1=Ispair(UU,UU) ==>\ |
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28 \ (p1<<p2) = (Isfst(p1)<<Isfst(p2) & Issnd(p1)<<Issnd(p2))" |
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29 (fn prems => |
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30 [ |
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31 (cut_facts_tac prems 1), |
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32 (rtac (inst_sprod_po RS ssubst) 1), |
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33 (etac less_sprod1b 1) |
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34 ]); |
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35 |
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36 val less_sprod4b = prove_goal Sprod2.thy |
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37 "p << Ispair(UU,UU) ==> p = Ispair(UU,UU)" |
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38 (fn prems => |
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39 [ |
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40 (cut_facts_tac prems 1), |
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41 (rtac less_sprod2b 1), |
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42 (etac (inst_sprod_po RS subst) 1) |
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43 ]); |
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44 |
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45 val less_sprod4a = (less_sprod4b RS defined_Ispair_rev); |
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46 (* Ispair(?a,?b) << Ispair(UU,UU) ==> ?a = UU | ?b = UU *) |
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47 |
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48 val less_sprod4c = prove_goal Sprod2.thy |
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49 "[|Ispair(xa,ya)<<Ispair(x,y);~xa=UU;~ya=UU;~x=UU;~y=UU|] ==>\ |
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50 \ xa<<x & ya << y" |
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51 (fn prems => |
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52 [ |
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53 (cut_facts_tac prems 1), |
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54 (rtac less_sprod2c 1), |
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55 (etac (inst_sprod_po RS subst) 1), |
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56 (REPEAT (atac 1)) |
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57 ]); |
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58 |
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59 (* ------------------------------------------------------------------------ *) |
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60 (* type sprod is pointed *) |
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61 (* ------------------------------------------------------------------------ *) |
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62 |
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63 val minimal_sprod = prove_goal Sprod2.thy "Ispair(UU,UU)<<p" |
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64 (fn prems => |
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65 [ |
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66 (rtac less_sprod3a 1), |
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67 (rtac refl 1) |
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68 ]); |
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69 |
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70 (* ------------------------------------------------------------------------ *) |
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71 (* Ispair is monotone in both arguments *) |
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72 (* ------------------------------------------------------------------------ *) |
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73 |
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74 val monofun_Ispair1 = prove_goalw Sprod2.thy [monofun] "monofun(Ispair)" |
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75 (fn prems => |
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76 [ |
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77 (strip_tac 1), |
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78 (rtac (less_fun RS iffD2) 1), |
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79 (strip_tac 1), |
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80 (res_inst_tac [("Q", |
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81 " Ispair(y,xa) =Ispair(UU,UU)")] (excluded_middle RS disjE) 1), |
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82 (res_inst_tac [("Q", |
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83 " Ispair(x,xa) =Ispair(UU,UU)")] (excluded_middle RS disjE) 1), |
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84 (rtac (less_sprod3b RS iffD2) 1), |
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85 (atac 1), |
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86 (rtac conjI 1), |
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87 (rtac (Isfst RS ssubst) 1), |
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88 (etac (strict_Ispair_rev RS conjunct1) 1), |
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89 (etac (strict_Ispair_rev RS conjunct2) 1), |
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90 (rtac (Isfst RS ssubst) 1), |
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91 (etac (strict_Ispair_rev RS conjunct1) 1), |
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92 (etac (strict_Ispair_rev RS conjunct2) 1), |
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93 (atac 1), |
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94 (rtac (Issnd RS ssubst) 1), |
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95 (etac (strict_Ispair_rev RS conjunct1) 1), |
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96 (etac (strict_Ispair_rev RS conjunct2) 1), |
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97 (rtac (Issnd RS ssubst) 1), |
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98 (etac (strict_Ispair_rev RS conjunct1) 1), |
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99 (etac (strict_Ispair_rev RS conjunct2) 1), |
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100 (rtac refl_less 1), |
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101 (etac less_sprod3a 1), |
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102 (res_inst_tac [("Q", |
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103 " Ispair(x,xa) =Ispair(UU,UU)")] (excluded_middle RS disjE) 1), |
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104 (etac less_sprod3a 2), |
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105 (res_inst_tac [("P","Ispair(y,xa) = Ispair(UU,UU)")] notE 1), |
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106 (atac 2), |
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107 (rtac defined_Ispair 1), |
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108 (etac notUU_I 1), |
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109 (etac (strict_Ispair_rev RS conjunct1) 1), |
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110 (etac (strict_Ispair_rev RS conjunct2) 1) |
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111 ]); |
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112 |
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113 |
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114 val monofun_Ispair2 = prove_goalw Sprod2.thy [monofun] "monofun(Ispair(x))" |
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115 (fn prems => |
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116 [ |
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117 (strip_tac 1), |
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118 (res_inst_tac [("Q", |
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119 " Ispair(x,y) =Ispair(UU,UU)")] (excluded_middle RS disjE) 1), |
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120 (res_inst_tac [("Q", |
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121 " Ispair(x,xa) =Ispair(UU,UU)")] (excluded_middle RS disjE) 1), |
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122 (rtac (less_sprod3b RS iffD2) 1), |
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123 (atac 1), |
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124 (rtac conjI 1), |
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125 (rtac (Isfst RS ssubst) 1), |
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126 (etac (strict_Ispair_rev RS conjunct1) 1), |
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127 (etac (strict_Ispair_rev RS conjunct2) 1), |
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128 (rtac (Isfst RS ssubst) 1), |
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129 (etac (strict_Ispair_rev RS conjunct1) 1), |
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130 (etac (strict_Ispair_rev RS conjunct2) 1), |
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131 (rtac refl_less 1), |
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132 (rtac (Issnd RS ssubst) 1), |
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133 (etac (strict_Ispair_rev RS conjunct1) 1), |
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134 (etac (strict_Ispair_rev RS conjunct2) 1), |
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135 (rtac (Issnd RS ssubst) 1), |
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136 (etac (strict_Ispair_rev RS conjunct1) 1), |
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137 (etac (strict_Ispair_rev RS conjunct2) 1), |
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138 (atac 1), |
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139 (etac less_sprod3a 1), |
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140 (res_inst_tac [("Q", |
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141 " Ispair(x,xa) =Ispair(UU,UU)")] (excluded_middle RS disjE) 1), |
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142 (etac less_sprod3a 2), |
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143 (res_inst_tac [("P","Ispair(x,y) = Ispair(UU,UU)")] notE 1), |
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144 (atac 2), |
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145 (rtac defined_Ispair 1), |
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146 (etac (strict_Ispair_rev RS conjunct1) 1), |
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147 (etac notUU_I 1), |
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148 (etac (strict_Ispair_rev RS conjunct2) 1) |
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149 ]); |
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150 |
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151 val monofun_Ispair = prove_goal Sprod2.thy |
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152 "[|x1<<x2; y1<<y2|] ==> Ispair(x1,y1)<<Ispair(x2,y2)" |
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153 (fn prems => |
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154 [ |
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155 (cut_facts_tac prems 1), |
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156 (rtac trans_less 1), |
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157 (rtac (monofun_Ispair1 RS monofunE RS spec RS spec RS mp RS |
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158 (less_fun RS iffD1 RS spec)) 1), |
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159 (rtac (monofun_Ispair2 RS monofunE RS spec RS spec RS mp) 2), |
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160 (atac 1), |
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161 (atac 1) |
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162 ]); |
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163 |
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164 |
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165 (* ------------------------------------------------------------------------ *) |
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166 (* Isfst and Issnd are monotone *) |
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167 (* ------------------------------------------------------------------------ *) |
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168 |
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169 val monofun_Isfst = prove_goalw Sprod2.thy [monofun] "monofun(Isfst)" |
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170 (fn prems => |
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171 [ |
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172 (strip_tac 1), |
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173 (res_inst_tac [("p","x")] IsprodE 1), |
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174 (hyp_subst_tac 1), |
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175 (rtac trans_less 1), |
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176 (rtac minimal 2), |
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177 (rtac (strict_Isfst1 RS ssubst) 1), |
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178 (rtac refl_less 1), |
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179 (hyp_subst_tac 1), |
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180 (res_inst_tac [("p","y")] IsprodE 1), |
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181 (hyp_subst_tac 1), |
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182 (res_inst_tac [("t","Isfst(Ispair(xa,ya))")] subst 1), |
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183 (rtac refl_less 2), |
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184 (etac (less_sprod4b RS sym RS arg_cong) 1), |
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185 (hyp_subst_tac 1), |
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186 (rtac (Isfst RS ssubst) 1), |
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187 (atac 1), |
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188 (atac 1), |
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189 (rtac (Isfst RS ssubst) 1), |
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190 (atac 1), |
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191 (atac 1), |
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192 (etac (less_sprod4c RS conjunct1) 1), |
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193 (REPEAT (atac 1)) |
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194 ]); |
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195 |
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196 val monofun_Issnd = prove_goalw Sprod2.thy [monofun] "monofun(Issnd)" |
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197 (fn prems => |
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198 [ |
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199 (strip_tac 1), |
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200 (res_inst_tac [("p","x")] IsprodE 1), |
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201 (hyp_subst_tac 1), |
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202 (rtac trans_less 1), |
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203 (rtac minimal 2), |
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204 (rtac (strict_Issnd1 RS ssubst) 1), |
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205 (rtac refl_less 1), |
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206 (hyp_subst_tac 1), |
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207 (res_inst_tac [("p","y")] IsprodE 1), |
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208 (hyp_subst_tac 1), |
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209 (res_inst_tac [("t","Issnd(Ispair(xa,ya))")] subst 1), |
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210 (rtac refl_less 2), |
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211 (etac (less_sprod4b RS sym RS arg_cong) 1), |
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212 (hyp_subst_tac 1), |
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213 (rtac (Issnd RS ssubst) 1), |
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214 (atac 1), |
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215 (atac 1), |
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216 (rtac (Issnd RS ssubst) 1), |
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217 (atac 1), |
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218 (atac 1), |
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219 (etac (less_sprod4c RS conjunct2) 1), |
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220 (REPEAT (atac 1)) |
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221 ]); |
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222 |
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223 |
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224 (* ------------------------------------------------------------------------ *) |
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225 (* the type 'a ** 'b is a cpo *) |
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226 (* ------------------------------------------------------------------------ *) |
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227 |
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228 val lub_sprod = prove_goal Sprod2.thy |
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229 "[|is_chain(S)|] ==> range(S) <<| \ |
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230 \ Ispair(lub(range(%i.Isfst(S(i)))),lub(range(%i.Issnd(S(i)))))" |
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231 (fn prems => |
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232 [ |
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233 (cut_facts_tac prems 1), |
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234 (rtac is_lubI 1), |
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235 (rtac conjI 1), |
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236 (rtac ub_rangeI 1), |
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237 (rtac allI 1), |
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238 (res_inst_tac [("t","S(i)")] (surjective_pairing_Sprod RS ssubst) 1), |
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239 (rtac monofun_Ispair 1), |
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240 (rtac is_ub_thelub 1), |
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241 (etac (monofun_Isfst RS ch2ch_monofun) 1), |
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242 (rtac is_ub_thelub 1), |
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243 (etac (monofun_Issnd RS ch2ch_monofun) 1), |
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244 (strip_tac 1), |
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245 (res_inst_tac [("t","u")] (surjective_pairing_Sprod RS ssubst) 1), |
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246 (rtac monofun_Ispair 1), |
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247 (rtac is_lub_thelub 1), |
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248 (etac (monofun_Isfst RS ch2ch_monofun) 1), |
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249 (etac (monofun_Isfst RS ub2ub_monofun) 1), |
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250 (rtac is_lub_thelub 1), |
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251 (etac (monofun_Issnd RS ch2ch_monofun) 1), |
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252 (etac (monofun_Issnd RS ub2ub_monofun) 1) |
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253 ]); |
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254 |
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255 val thelub_sprod = (lub_sprod RS thelubI); |
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256 (* is_chain(?S1) ==> lub(range(?S1)) = *) |
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257 (* Ispair(lub(range(%i. Isfst(?S1(i)))),lub(range(%i. Issnd(?S1(i))))) *) |
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258 |
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259 val cpo_sprod = prove_goal Sprod2.thy |
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260 "is_chain(S::nat=>'a**'b)==>? x.range(S)<<| x" |
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261 (fn prems => |
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262 [ |
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263 (cut_facts_tac prems 1), |
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264 (rtac exI 1), |
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265 (etac lub_sprod 1) |
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266 ]); |
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267 |
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268 |
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269 |
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270 |
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271 |
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272 |
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273 |
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274 |