1 (*<*) |
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2 theory Slides1 |
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3 imports "~~/src/HOL/Library/LaTeXsugar" "Nominal" |
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4 begin |
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5 |
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6 notation (latex output) |
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7 set ("_") and |
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8 Cons ("_::/_" [66,65] 65) |
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9 |
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10 (*>*) |
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11 |
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12 |
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13 text_raw {* |
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14 %%\renewcommand{\slidecaption}{Cambridge, 8.~June 2010} |
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15 \renewcommand{\slidecaption}{Uppsala, 3.~March 2011} |
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16 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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17 \mode<presentation>{ |
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18 \begin{frame}<1>[t] |
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19 \frametitle{% |
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20 \begin{tabular}{@ {\hspace{-3mm}}c@ {}} |
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21 \\ |
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22 \huge Nominal Isabelle 2\\[-2mm] |
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23 \large Or, How to Reason Conveniently\\[-5mm] |
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24 \large with General Bindings\\[5mm] |
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25 \end{tabular}} |
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26 \begin{center} |
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27 Christian Urban |
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28 \end{center} |
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29 \begin{center} |
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30 joint work with {\bf Cezary Kaliszyk}\\[0mm] |
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31 \end{center} |
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32 \end{frame}} |
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33 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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34 |
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35 *} |
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36 |
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37 text_raw {* |
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38 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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39 \mode<presentation>{ |
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40 \begin{frame}<1-2> |
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41 \frametitle{\begin{tabular}{c}Binding in Old Nominal\end{tabular}} |
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42 \mbox{}\\[-6mm] |
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43 |
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44 \begin{itemize} |
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45 \item the old Nominal Isabelle provided a reasoning infrastructure for single binders\medskip |
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46 |
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47 \begin{center} |
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48 Lam [a].(Var a) |
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49 \end{center}\bigskip |
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50 |
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51 \item<2-> but representing |
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52 |
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53 \begin{center} |
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54 $\forall\{a_1,\ldots,a_n\}.\; T$ |
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55 \end{center}\medskip |
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56 |
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57 with single binders and reasoning about it is a \alert{\bf major} pain; |
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58 take my word for it! |
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59 \end{itemize} |
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60 |
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61 \only<1>{ |
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62 \begin{textblock}{6}(1.5,11) |
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63 \small |
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64 for example\\ |
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65 \begin{tabular}{l@ {\hspace{2mm}}l} |
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66 & a $\fresh$ Lam [a]. t\\ |
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67 & Lam [a]. (Var a) \alert{$=$} Lam [b]. (Var b)\\ |
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68 & Barendregt-style reasoning about bound variables\\ |
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69 \end{tabular} |
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70 \end{textblock}} |
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71 |
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72 \end{frame}} |
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73 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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74 *} |
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75 |
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76 text_raw {* |
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77 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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78 \mode<presentation>{ |
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79 \begin{frame}<1-4> |
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80 \frametitle{\begin{tabular}{c}Binding Sets of Names\end{tabular}} |
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81 \mbox{}\\[-3mm] |
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82 |
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83 \begin{itemize} |
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84 \item binding sets of names has some interesting properties:\medskip |
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85 |
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86 \begin{center} |
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87 \begin{tabular}{l} |
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88 $\forall\{x, y\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{y, x\}.\, y \rightarrow x$ |
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89 \bigskip\smallskip\\ |
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90 |
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91 \onslide<2->{% |
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92 $\forall\{x, y\}.\, x \rightarrow y \;\;\not\approx_\alpha\;\; \forall\{z\}.\, z \rightarrow z$ |
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93 }\bigskip\smallskip\\ |
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94 |
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95 \onslide<3->{% |
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96 $\forall\{x\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{x, \alert{z}\}.\, x \rightarrow y$ |
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97 }\medskip\\ |
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98 \onslide<3->{\hspace{4cm}\small provided $z$ is fresh for the type} |
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99 \end{tabular} |
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100 \end{center} |
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101 \end{itemize} |
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102 |
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103 \begin{textblock}{8}(2,14.5) |
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104 \footnotesize $^*$ $x$, $y$, $z$ are assumed to be distinct |
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105 \end{textblock} |
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106 |
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107 \only<4>{ |
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108 \begin{textblock}{6}(2.5,4) |
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109 \begin{tikzpicture} |
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110 \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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111 {\normalsize\color{darkgray} |
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112 \begin{minipage}{8cm}\raggedright |
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113 For type-schemes the order of bound names does not matter, and |
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114 alpha-equivalence is preserved under \alert{vacuous} binders. |
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115 \end{minipage}}; |
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116 \end{tikzpicture} |
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117 \end{textblock}} |
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118 \end{frame}} |
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119 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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120 *} |
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121 |
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122 text_raw {* |
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123 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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124 \mode<presentation>{ |
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125 \begin{frame}<1-3> |
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126 \frametitle{\begin{tabular}{c}Other Binding Modes\end{tabular}} |
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127 \mbox{}\\[-3mm] |
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128 |
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129 \begin{itemize} |
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130 \item alpha-equivalence being preserved under vacuous binders is \underline{not} always |
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131 wanted:\bigskip\bigskip\normalsize |
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132 |
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133 \begin{tabular}{@ {\hspace{-8mm}}l} |
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134 $\text{let}\;x = 3\;\text{and}\;y = 2\;\text{in}\;x - y\;\text{end}$\medskip\\ |
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135 \onslide<2->{$\;\;\;\only<2>{\approx_\alpha}\only<3>{\alert{\not\approx_\alpha}} |
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136 \text{let}\;y = 2\;\text{and}\;x = 3\only<3->{\alert{\;\text{and} |
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137 \;z = \text{loop}}}\;\text{in}\;x - y\;\text{end}$} |
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138 \end{tabular} |
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139 |
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140 |
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141 \end{itemize} |
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142 |
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143 \end{frame}} |
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144 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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145 *} |
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146 |
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147 text_raw {* |
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148 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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149 \mode<presentation>{ |
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150 \begin{frame}<1> |
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151 \frametitle{\begin{tabular}{c}\LARGE{}Even Another Binding Mode\end{tabular}} |
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152 \mbox{}\\[-3mm] |
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153 |
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154 \begin{itemize} |
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155 \item sometimes one wants to abstract more than one name, but the order \underline{does} matter\bigskip |
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156 |
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157 \begin{center} |
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158 \begin{tabular}{@ {\hspace{-8mm}}l} |
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159 $\text{let}\;(x, y) = (3, 2)\;\text{in}\;x - y\;\text{end}$\medskip\\ |
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160 $\;\;\;\not\approx_\alpha |
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161 \text{let}\;(y, x) = (3, 2)\;\text{in}\;x - y\;\text{end}$ |
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162 \end{tabular} |
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163 \end{center} |
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164 |
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165 |
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166 \end{itemize} |
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167 |
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168 \end{frame}} |
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169 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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170 *} |
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171 |
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172 text_raw {* |
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173 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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174 \mode<presentation>{ |
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175 \begin{frame}<1-2> |
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176 \frametitle{\begin{tabular}{c}\LARGE{}Three Binding Modes\end{tabular}} |
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177 \mbox{}\\[-3mm] |
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178 |
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179 \begin{itemize} |
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180 \item the order does not matter and alpha-equivelence is preserved under |
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181 vacuous binders \textcolor{gray}{(restriction)}\medskip |
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182 |
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183 \item the order does not matter, but the cardinality of the binders |
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184 must be the same \textcolor{gray}{(abstraction)}\medskip |
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185 |
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186 \item the order does matter \textcolor{gray}{(iterated single binders)} |
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187 \end{itemize} |
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188 |
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189 \onslide<2->{ |
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190 \begin{center} |
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191 \isacommand{bind (set+)}\hspace{6mm} |
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192 \isacommand{bind (set)}\hspace{6mm} |
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193 \isacommand{bind} |
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194 \end{center}} |
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195 |
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196 \end{frame}} |
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197 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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198 *} |
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199 |
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200 text_raw {* |
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201 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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202 \mode<presentation>{ |
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203 \begin{frame}<1-3> |
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204 \frametitle{\begin{tabular}{c}Specification of Binding\end{tabular}} |
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205 \mbox{}\\[-6mm] |
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206 |
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207 \mbox{}\hspace{10mm} |
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208 \begin{tabular}{ll} |
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209 \multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\ |
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210 \hspace{5mm}\phantom{$|$} Var name\\ |
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211 \hspace{5mm}$|$ App trm trm\\ |
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212 \hspace{5mm}$|$ Lam \only<2->{x::}name \only<2->{t::}trm |
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213 & \onslide<2->{\isacommand{bind} x \isacommand{in} t}\\ |
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214 \hspace{5mm}$|$ Let \only<2->{as::}assn \only<2->{t::}trm |
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215 & \onslide<2->{\isacommand{bind} bn(as) \isacommand{in} t}\\ |
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216 \multicolumn{2}{l}{\isacommand{and} assn $=$}\\ |
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217 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\ |
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218 \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\ |
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219 \multicolumn{2}{l}{\onslide<3->{\isacommand{binder} bn \isacommand{where}}}\\ |
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220 \multicolumn{2}{l}{\onslide<3->{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ []}}\\ |
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221 \multicolumn{2}{l}{\onslide<3->{\hspace{5mm}$|$ bn(ACons a t as) $=$ [a] @ bn(as)}}\\ |
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222 \end{tabular} |
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223 |
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224 |
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225 |
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226 \end{frame}} |
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227 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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228 *} |
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229 |
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230 text_raw {* |
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231 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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232 \mode<presentation>{ |
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233 \begin{frame}<1-5> |
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234 \frametitle{\begin{tabular}{c}Inspiration from Ott\end{tabular}} |
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235 \mbox{}\\[-3mm] |
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236 |
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237 \begin{itemize} |
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238 \item this way of specifying binding is inspired by |
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239 {\bf Ott}\onslide<2->{, \alert{\bf but} we made some adjustments:}\medskip |
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240 |
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241 |
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242 \only<2>{ |
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243 \begin{itemize} |
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244 \item Ott allows specifications like\smallskip |
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245 \begin{center} |
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246 $t ::= t\;t\; |\;\lambda x.t$ |
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247 \end{center} |
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248 \end{itemize}} |
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249 |
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250 \only<3-4>{ |
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251 \begin{itemize} |
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252 \item whether something is bound can depend in Ott on other bound things\smallskip |
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253 \begin{center} |
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254 \begin{tikzpicture} |
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255 \node (A) at (-0.5,1) {Foo $(\lambda y. \lambda x. t)$}; |
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256 \node (B) at ( 1.1,1) {$s$}; |
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257 \onslide<4>{\node (C) at (0.5,0) {$\{y, x\}$};} |
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258 \onslide<4>{\draw[->,red,line width=1mm] (A) -- (C);} |
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259 \onslide<4>{\draw[->,red,line width=1mm] (C) -- (B);} |
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260 \end{tikzpicture} |
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261 \end{center} |
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262 \onslide<4>{this might make sense for ``raw'' terms, but not at all |
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263 for $\alpha$-equated terms} |
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264 \end{itemize}} |
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265 |
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266 \only<5>{ |
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267 \begin{itemize} |
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268 \item we allow multiple ``binders'' and ``bodies''\smallskip |
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269 \begin{center} |
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270 \begin{tabular}{l} |
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271 \isacommand{bind} a b c \ldots \isacommand{in} x y z \ldots\\ |
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272 \isacommand{bind (set)} a b c \ldots \isacommand{in} x y z \ldots\\ |
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273 \isacommand{bind (set+)} a b c \ldots \isacommand{in} x y z \ldots |
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274 \end{tabular} |
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275 \end{center}\bigskip\medskip |
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276 the reason is that with our definition of $\alpha$-equivalence\medskip |
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277 \begin{center} |
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278 \begin{tabular}{l} |
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279 \isacommand{bind (set+)} as \isacommand{in} x y $\not\Leftrightarrow$\\ |
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280 \hspace{8mm}\isacommand{bind (set+)} as \isacommand{in} x, \isacommand{bind (set+)} as \isacommand{in} y |
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281 \end{tabular} |
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282 \end{center}\medskip |
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283 |
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284 same with \isacommand{bind (set)} |
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285 \end{itemize}} |
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286 \end{itemize} |
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287 |
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288 |
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289 \end{frame}} |
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290 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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291 *} |
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292 |
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293 text_raw {* |
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294 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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295 \mode<presentation>{ |
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296 \begin{frame}<1> |
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297 \frametitle{\begin{tabular}{c}Alpha-Equivalence\end{tabular}} |
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298 \mbox{}\\[-3mm] |
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299 |
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300 \begin{itemize} |
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301 \item in the old Nominal Isabelle, we represented single binders as partial functions:\bigskip |
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302 |
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303 \begin{center} |
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304 \begin{tabular}{l} |
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305 Lam [$a$].\,$t$ $\;{^\text{``}}\!\dn{}\!^{\text{''}}$\\[2mm] |
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306 \;\;\;\;$\lambda b.$\;$\text{if}\;a = b\;\text{then}\;t\;\text{else}$\\ |
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307 \phantom{\;\;\;\;$\lambda b.$\;\;\;\;\;\;}$\text{if}\;b \fresh t\; |
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308 \text{then}\;(a\;b)\act t\;\text{else}\;\text{error}$ |
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309 \end{tabular} |
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310 \end{center} |
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311 \end{itemize} |
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312 |
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313 \begin{textblock}{10}(2,14) |
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314 \footnotesize $^*$ alpha-equality coincides with equality on functions |
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315 \end{textblock} |
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316 \end{frame}} |
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317 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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318 *} |
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319 |
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320 text_raw {* |
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321 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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322 \mode<presentation>{ |
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323 \begin{frame}<1-> |
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324 \frametitle{\begin{tabular}{c}New Design\end{tabular}} |
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325 \mbox{}\\[4mm] |
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326 |
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327 \begin{center} |
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328 \begin{tikzpicture} |
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329 {\draw (0,0) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm] |
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330 (A) {\begin{minipage}{1.1cm}bind.\\spec.\end{minipage}};} |
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331 |
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332 {\draw (3,0) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm] |
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333 (B) {\begin{minipage}{1.1cm}raw\\terms\end{minipage}};} |
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334 |
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335 \alt<2> |
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336 {\draw (6,0) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm] |
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337 (C) {\textcolor{red}{\begin{minipage}{1.1cm}$\alpha$-\\equiv.\end{minipage}}};} |
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338 {\draw (6,0) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm] |
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339 (C) {\begin{minipage}{1.1cm}$\alpha$-\\equiv.\end{minipage}};} |
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340 |
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341 {\draw (0,-3) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm] |
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342 (D) {\begin{minipage}{1.1cm}quot.\\type\end{minipage}};} |
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343 |
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344 {\draw (3,-3) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm] |
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345 (E) {\begin{minipage}{1.1cm}lift\\thms\end{minipage}};} |
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346 |
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347 {\draw (6,-3) node[inner sep=3mm, ultra thick, draw=fg, rounded corners=2mm] |
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348 (F) {\begin{minipage}{1.1cm}add.\\thms\end{minipage}};} |
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349 |
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350 \draw[->,fg!50,line width=1mm] (A) -- (B); |
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351 \draw[->,fg!50,line width=1mm] (B) -- (C); |
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352 \draw[->,fg!50,line width=1mm, line join=round, rounded corners=2mm] |
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353 (C) -- (8,0) -- (8,-1.5) -- (-2,-1.5) -- (-2,-3) -- (D); |
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354 \draw[->,fg!50,line width=1mm] (D) -- (E); |
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355 \draw[->,fg!50,line width=1mm] (E) -- (F); |
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356 \end{tikzpicture} |
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357 \end{center} |
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358 |
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359 \end{frame}} |
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360 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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361 *} |
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362 |
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363 |
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364 |
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365 text_raw {* |
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366 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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367 \mode<presentation>{ |
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368 \begin{frame}<1-8> |
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369 \frametitle{\begin{tabular}{c}Alpha-Equivalence\end{tabular}} |
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370 \mbox{}\\[-3mm] |
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371 |
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372 \begin{itemize} |
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373 \item lets first look at pairs\bigskip\medskip |
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374 |
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375 \begin{tabular}{@ {\hspace{1cm}}l} |
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376 $(as, x) \onslide<2->{\approx\!}\makebox[0mm][l]{\only<2-6>{${}_{\text{set}}$}% |
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377 \only<7>{${}_{\text{\alert{list}}}$}% |
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378 \only<8>{${}_{\text{\alert{set+}}}$}}% |
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379 \onslide<3->{^{R,\text{fv}}}\,\onslide<2->{(bs,y)}$ |
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380 \end{tabular}\bigskip |
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381 \end{itemize} |
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382 |
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383 \only<1>{ |
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384 \begin{textblock}{8}(3,8.5) |
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385 \begin{tabular}{l@ {\hspace{2mm}}p{8cm}} |
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386 & $as$ is a set of names\ldots the binders\\ |
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387 & $x$ is the body (might be a tuple)\\ |
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388 & $\approx_{\text{set}}$ is where the cardinality |
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389 of the binders has to be the same\\ |
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390 \end{tabular} |
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391 \end{textblock}} |
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392 |
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393 \only<4->{ |
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394 \begin{textblock}{12}(5,8) |
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395 \begin{tabular}{ll@ {\hspace{1mm}}l} |
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396 $\dn$ & \onslide<5->{$\exists \pi.\,$} & $\text{fv}(x) - as = \text{fv}(y) - bs$\\[1mm] |
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397 & \onslide<5->{$\;\;\;\wedge$} & \onslide<5->{$\text{fv}(x) - as \fresh^* \pi$}\\[1mm] |
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398 & \onslide<5->{$\;\;\;\wedge$} & \onslide<5->{$(\pi \act x)\;R\;y$}\\[1mm] |
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399 & \onslide<6-7>{$\;\;\;\wedge$} & \onslide<6-7>{$\pi \act as = bs$}\\ |
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400 \end{tabular} |
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401 \end{textblock}} |
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402 |
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403 \only<7>{ |
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404 \begin{textblock}{7}(3,13.8) |
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405 \footnotesize $^*$ $as$ and $bs$ are \alert{lists} of names |
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406 \end{textblock}} |
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407 \end{frame}} |
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408 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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409 *} |
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410 |
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411 text_raw {* |
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412 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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413 \mode<presentation>{ |
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414 \begin{frame}<1-3> |
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415 \frametitle{\begin{tabular}{c}Examples\end{tabular}} |
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416 \mbox{}\\[-3mm] |
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417 |
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418 \begin{itemize} |
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419 \item lets look at ``type-schemes'':\medskip\medskip |
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420 |
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421 \begin{center} |
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422 $(as, x) \approx\!\makebox[0mm][l]{${}_{\text{set}}$}\only<1>{{}^{R,\text{fv}}}\only<2->{{}^{\alert{=},\alert{\text{fv}}}} (bs, y)$ |
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423 \end{center}\medskip |
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424 |
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425 \onslide<2->{ |
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426 \begin{center} |
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427 \begin{tabular}{l} |
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428 $\text{fv}(x) = \{x\}$\\[1mm] |
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429 $\text{fv}(T_1 \rightarrow T_2) = \text{fv}(T_1) \cup \text{fv}(T_2)$\\ |
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430 \end{tabular} |
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431 \end{center}} |
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432 \end{itemize} |
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433 |
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434 |
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435 \only<3->{ |
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436 \begin{textblock}{4}(0.3,12) |
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437 \begin{tikzpicture} |
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438 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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439 {\tiny\color{darkgray} |
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440 \begin{minipage}{3.4cm}\raggedright |
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441 \begin{tabular}{r@ {\hspace{1mm}}l} |
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442 \multicolumn{2}{@ {}l}{set+:}\\ |
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443 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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444 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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445 $\wedge$ & $\pi \cdot x = y$\\ |
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446 \\ |
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447 \end{tabular} |
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448 \end{minipage}}; |
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449 \end{tikzpicture} |
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450 \end{textblock}} |
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451 \only<3->{ |
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452 \begin{textblock}{4}(5.2,12) |
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453 \begin{tikzpicture} |
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454 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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455 {\tiny\color{darkgray} |
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456 \begin{minipage}{3.4cm}\raggedright |
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457 \begin{tabular}{r@ {\hspace{1mm}}l} |
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458 \multicolumn{2}{@ {}l}{set:}\\ |
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459 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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460 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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461 $\wedge$ & $\pi \cdot x = y$\\ |
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462 $\wedge$ & $\pi \cdot as = bs$\\ |
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463 \end{tabular} |
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464 \end{minipage}}; |
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465 \end{tikzpicture} |
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466 \end{textblock}} |
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467 \only<3->{ |
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468 \begin{textblock}{4}(10.2,12) |
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469 \begin{tikzpicture} |
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470 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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471 {\tiny\color{darkgray} |
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472 \begin{minipage}{3.4cm}\raggedright |
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473 \begin{tabular}{r@ {\hspace{1mm}}l} |
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474 \multicolumn{2}{@ {}l}{list:}\\ |
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475 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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476 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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477 $\wedge$ & $\pi \cdot x = y$\\ |
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478 $\wedge$ & $\pi \cdot as = bs$\\ |
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479 \end{tabular} |
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480 \end{minipage}}; |
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481 \end{tikzpicture} |
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482 \end{textblock}} |
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483 |
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484 \end{frame}} |
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485 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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486 *} |
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487 |
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488 text_raw {* |
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489 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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490 \mode<presentation>{ |
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491 \begin{frame}<1-2> |
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492 \frametitle{\begin{tabular}{c}Examples\end{tabular}} |
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493 \mbox{}\\[-3mm] |
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494 |
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495 \begin{center} |
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496 \only<1>{$(\{x, y\}, x \rightarrow y) \approx_? (\{x, y\}, y \rightarrow x)$} |
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497 \only<2>{$([x, y], x \rightarrow y) \approx_? ([x, y], y \rightarrow x)$} |
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498 \end{center} |
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499 |
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500 \begin{itemize} |
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501 \item $\approx_{\text{set+}}$, $\approx_{\text{set}}$% |
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502 \only<2>{, \alert{$\not\approx_{\text{list}}$}} |
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503 \end{itemize} |
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504 |
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505 |
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506 \only<1->{ |
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507 \begin{textblock}{4}(0.3,12) |
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508 \begin{tikzpicture} |
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509 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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510 {\tiny\color{darkgray} |
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511 \begin{minipage}{3.4cm}\raggedright |
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512 \begin{tabular}{r@ {\hspace{1mm}}l} |
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513 \multicolumn{2}{@ {}l}{set+:}\\ |
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514 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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515 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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516 $\wedge$ & $\pi \cdot x = y$\\ |
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517 \\ |
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518 \end{tabular} |
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519 \end{minipage}}; |
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520 \end{tikzpicture} |
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521 \end{textblock}} |
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522 \only<1->{ |
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523 \begin{textblock}{4}(5.2,12) |
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524 \begin{tikzpicture} |
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525 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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526 {\tiny\color{darkgray} |
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527 \begin{minipage}{3.4cm}\raggedright |
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528 \begin{tabular}{r@ {\hspace{1mm}}l} |
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529 \multicolumn{2}{@ {}l}{set:}\\ |
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530 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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531 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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532 $\wedge$ & $\pi \cdot x = y$\\ |
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533 $\wedge$ & $\pi \cdot as = bs$\\ |
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534 \end{tabular} |
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535 \end{minipage}}; |
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536 \end{tikzpicture} |
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537 \end{textblock}} |
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538 \only<1->{ |
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539 \begin{textblock}{4}(10.2,12) |
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540 \begin{tikzpicture} |
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541 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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542 {\tiny\color{darkgray} |
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543 \begin{minipage}{3.4cm}\raggedright |
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544 \begin{tabular}{r@ {\hspace{1mm}}l} |
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545 \multicolumn{2}{@ {}l}{list:}\\ |
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546 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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547 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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548 $\wedge$ & $\pi \cdot x = y$\\ |
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549 $\wedge$ & $\pi \cdot as = bs$\\ |
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550 \end{tabular} |
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551 \end{minipage}}; |
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552 \end{tikzpicture} |
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553 \end{textblock}} |
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554 |
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555 \end{frame}} |
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556 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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557 *} |
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558 |
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559 text_raw {* |
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560 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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561 \mode<presentation>{ |
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562 \begin{frame}<1-2> |
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563 \frametitle{\begin{tabular}{c}Examples\end{tabular}} |
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564 \mbox{}\\[-3mm] |
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565 |
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566 \begin{center} |
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567 \only<1>{$(\{x\}, x) \approx_? (\{x, y\}, x)$} |
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568 \end{center} |
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569 |
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570 \begin{itemize} |
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571 \item $\approx_{\text{set+}}$, $\not\approx_{\text{set}}$, |
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572 $\not\approx_{\text{list}}$ |
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573 \end{itemize} |
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574 |
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575 |
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576 \only<1->{ |
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577 \begin{textblock}{4}(0.3,12) |
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578 \begin{tikzpicture} |
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579 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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580 {\tiny\color{darkgray} |
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581 \begin{minipage}{3.4cm}\raggedright |
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582 \begin{tabular}{r@ {\hspace{1mm}}l} |
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583 \multicolumn{2}{@ {}l}{set+:}\\ |
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584 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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585 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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586 $\wedge$ & $\pi \cdot x = y$\\ |
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587 \\ |
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588 \end{tabular} |
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589 \end{minipage}}; |
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590 \end{tikzpicture} |
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591 \end{textblock}} |
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592 \only<1->{ |
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595 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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596 {\tiny\color{darkgray} |
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597 \begin{minipage}{3.4cm}\raggedright |
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598 \begin{tabular}{r@ {\hspace{1mm}}l} |
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599 \multicolumn{2}{@ {}l}{set:}\\ |
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600 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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601 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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602 $\wedge$ & $\pi \cdot x = y$\\ |
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603 $\wedge$ & $\pi \cdot as = bs$\\ |
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604 \end{tabular} |
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605 \end{minipage}}; |
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606 \end{tikzpicture} |
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607 \end{textblock}} |
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608 \only<1->{ |
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609 \begin{textblock}{4}(10.2,12) |
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610 \begin{tikzpicture} |
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611 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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612 {\tiny\color{darkgray} |
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613 \begin{minipage}{3.4cm}\raggedright |
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614 \begin{tabular}{r@ {\hspace{1mm}}l} |
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615 \multicolumn{2}{@ {}l}{list:}\\ |
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616 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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617 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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618 $\wedge$ & $\pi \cdot x = y$\\ |
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619 $\wedge$ & $\pi \cdot as = bs$\\ |
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620 \end{tabular} |
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621 \end{minipage}}; |
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622 \end{tikzpicture} |
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623 \end{textblock}} |
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624 |
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625 \only<2>{ |
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626 \begin{textblock}{6}(2.5,4) |
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627 \begin{tikzpicture} |
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628 \draw (0,0) node[inner sep=5mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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629 {\normalsize |
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630 \begin{minipage}{8cm}\raggedright |
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631 \begin{itemize} |
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632 \item \color{darkgray}$\alpha$-equivalences coincide when a single name is |
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633 abstracted |
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634 \item \color{darkgray}in that case they are equivalent to ``old-fashioned'' definitions of $\alpha$ |
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635 \end{itemize} |
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636 \end{minipage}}; |
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637 \end{tikzpicture} |
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638 \end{textblock}} |
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639 |
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640 \end{frame}} |
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641 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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642 *} |
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643 |
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644 text_raw {* |
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645 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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646 \mode<presentation>{ |
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647 \begin{frame}<1-3> |
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648 \frametitle{\begin{tabular}{c}General Abstractions\end{tabular}} |
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649 \mbox{}\\[-7mm] |
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650 |
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651 \begin{itemize} |
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652 \item we take $(as, x) \approx\!\makebox[0mm][l]{${}_{{}*{}}$}^{=,\text{supp}} (bs, y)$\medskip |
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653 \item they are equivalence relations\medskip |
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654 \item we can therefore use the quotient package to introduce the |
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655 types $\beta\;\text{abs}_*$\bigskip |
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656 \begin{center} |
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657 \only<1>{$[as].\,x$} |
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658 \only<2>{$\text{supp}([as].x) = \text{supp}(x) - as$} |
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659 \only<3>{% |
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660 \begin{tabular}{r@ {\hspace{1mm}}l} |
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661 \multicolumn{2}{@ {\hspace{-7mm}}l}{$[as]. x \alert{=} [bs].y\;\;\;\text{if\!f}$}\\[2mm] |
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662 $\exists \pi.$ & $\text{supp}(x) - as = \text{supp}(y) - bs$\\ |
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663 $\wedge$ & $\text{supp}(x) - as \fresh^* \pi$\\ |
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664 $\wedge$ & $\pi \act x = y $\\ |
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665 $(\wedge$ & $\pi \act as = bs)\;^*$\\ |
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666 \end{tabular}} |
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667 \end{center} |
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668 \end{itemize} |
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669 |
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670 \only<1->{ |
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671 \begin{textblock}{8}(12,3.8) |
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672 \footnotesize $^*$ set, set+, list |
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673 \end{textblock}} |
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674 |
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675 \end{frame}} |
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676 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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677 *} |
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678 |
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679 text_raw {* |
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680 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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681 \mode<presentation>{ |
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682 \begin{frame}<1> |
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683 \frametitle{\begin{tabular}{c}A Problem\end{tabular}} |
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684 \mbox{}\\[-3mm] |
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685 |
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686 \begin{center} |
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687 $\text{let}\;x_1=t_1 \ldots x_n=t_n\;\text{in}\;s$ |
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688 \end{center} |
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689 |
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690 \begin{itemize} |
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691 \item we cannot represent this as\medskip |
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692 \begin{center} |
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693 $\text{let}\;[x_1,\ldots,x_n]\alert{.}s\;\;[t_1,\ldots,t_n]$ |
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694 \end{center}\bigskip |
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695 |
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696 because\medskip |
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697 \begin{center} |
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698 $\text{let}\;[x].s\;\;[t_1,t_2]$ |
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699 \end{center} |
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700 \end{itemize} |
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701 |
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702 |
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703 \end{frame}} |
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704 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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705 *} |
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706 |
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707 text_raw {* |
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708 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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709 \mode<presentation>{ |
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710 \begin{frame}<1-> |
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711 \frametitle{\begin{tabular}{c}Our Specifications\end{tabular}} |
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712 \mbox{}\\[-6mm] |
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713 |
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714 \mbox{}\hspace{10mm} |
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715 \begin{tabular}{ll} |
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716 \multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\ |
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717 \hspace{5mm}\phantom{$|$} Var name\\ |
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718 \hspace{5mm}$|$ App trm trm\\ |
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719 \hspace{5mm}$|$ Lam x::name t::trm |
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720 & \isacommand{bind} x \isacommand{in} t\\ |
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721 \hspace{5mm}$|$ Let as::assn t::trm |
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722 & \isacommand{bind} bn(as) \isacommand{in} t\\ |
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723 \multicolumn{2}{l}{\isacommand{and} assn $=$}\\ |
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724 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\ |
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725 \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\ |
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726 \multicolumn{2}{l}{\isacommand{binder} bn \isacommand{where}}\\ |
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727 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\ |
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728 \multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\ |
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729 \end{tabular} |
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730 |
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731 \end{frame}} |
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732 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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733 *} |
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734 |
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735 text_raw {* |
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736 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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737 \mode<presentation>{ |
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738 \begin{frame}<1-2> |
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739 \frametitle{\begin{tabular}{c}``Raw'' Definitions\end{tabular}} |
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740 \mbox{}\\[-6mm] |
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741 |
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742 \mbox{}\hspace{10mm} |
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743 \begin{tabular}{ll} |
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744 \multicolumn{2}{l}{\isacommand{datatype} trm $=$}\\ |
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745 \hspace{5mm}\phantom{$|$} Var name\\ |
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746 \hspace{5mm}$|$ App trm trm\\ |
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747 \hspace{5mm}$|$ Lam name trm\\ |
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748 \hspace{5mm}$|$ Let assn trm\\ |
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749 \multicolumn{2}{l}{\isacommand{and} assn $=$}\\ |
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750 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\ |
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751 \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\[5mm] |
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752 \multicolumn{2}{l}{\isacommand{function} bn \isacommand{where}}\\ |
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753 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\ |
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754 \multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\ |
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755 \end{tabular} |
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756 |
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757 \only<2>{ |
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758 \begin{textblock}{5}(10,5) |
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759 $+$ \begin{tabular}{l}automatically\\ |
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760 generate fv's\end{tabular} |
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761 \end{textblock}} |
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762 \end{frame}} |
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763 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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764 *} |
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765 |
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766 text_raw {* |
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767 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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768 \mode<presentation>{ |
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769 \begin{frame}<1> |
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770 \frametitle{\begin{tabular}{c}\LARGE``Raw'' Alpha-Equivalence\end{tabular}} |
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771 \mbox{}\\[6mm] |
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772 |
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773 \begin{center} |
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774 Lam x::name t::trm \hspace{10mm}\isacommand{bind} x \isacommand{in} t\\ |
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775 \end{center} |
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776 |
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777 |
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778 \[ |
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779 \infer[\text{Lam-}\!\approx_\alpha] |
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780 {\text{Lam}\;x\;t \approx_\alpha \text{Lam}\;x'\;t'} |
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781 {([x], t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$} |
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782 ^{\approx_\alpha,\text{fv}} ([x'], t')} |
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783 \] |
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784 |
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785 |
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786 \end{frame}} |
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787 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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788 *} |
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789 |
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790 text_raw {* |
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791 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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792 \mode<presentation>{ |
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793 \begin{frame}<1> |
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794 \frametitle{\begin{tabular}{c}\LARGE``Raw'' Alpha-Equivalence\end{tabular}} |
|
795 \mbox{}\\[6mm] |
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796 |
|
797 \begin{center} |
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798 Lam x::name y::name t::trm s::trm \hspace{5mm}\isacommand{bind} x y \isacommand{in} t s\\ |
|
799 \end{center} |
|
800 |
|
801 |
|
802 \[ |
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803 \infer[\text{Lam-}\!\approx_\alpha] |
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804 {\text{Lam}\;x\;y\;t\;s \approx_\alpha \text{Lam}\;x'\;y'\;t'\;s'} |
|
805 {([x, y], (t, s)) \approx\!\makebox[0mm][l]{${}_{\text{list}}$} |
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806 ^{R, fv} ([x', y'], (t', s'))} |
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807 \] |
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808 |
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809 \footnotesize |
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810 where $R =\;\approx_\alpha\times\approx_\alpha$ and $fv = \text{fv}\cup\text{fv}$ |
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811 |
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812 \end{frame}} |
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813 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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814 *} |
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815 |
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816 text_raw {* |
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817 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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818 \mode<presentation>{ |
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819 \begin{frame}<1-2> |
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820 \frametitle{\begin{tabular}{c}\LARGE``Raw'' Alpha-Equivalence\end{tabular}} |
|
821 \mbox{}\\[6mm] |
|
822 |
|
823 \begin{center} |
|
824 Let as::assn t::trm \hspace{10mm}\isacommand{bind} bn(as) \isacommand{in} t\\ |
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825 \end{center} |
|
826 |
|
827 |
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828 \[ |
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829 \infer[\text{Let-}\!\approx_\alpha] |
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830 {\text{Let}\;as\;t \approx_\alpha \text{Let}\;as'\;t'} |
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831 {(\text{bn}(as), t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$} |
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832 ^{\approx_\alpha,\text{fv}} (\text{bn}(as'), t') & |
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833 \onslide<2->{as \approx_\alpha^{\text{bn}} as'}} |
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834 \]\bigskip |
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835 |
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836 |
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837 \onslide<1->{\small{}bn-function $\Rightarrow$ \alert{deep binders}} |
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838 \end{frame}} |
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839 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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840 *} |
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841 |
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842 |
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843 text_raw {* |
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844 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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845 \mode<presentation>{ |
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846 \begin{frame}<1-> |
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847 \frametitle{\begin{tabular}{c}\LARGE{}$\alpha$ for Binding Functions\end{tabular}} |
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848 \mbox{}\\[-6mm] |
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849 |
|
850 \mbox{}\hspace{10mm} |
|
851 \begin{tabular}{l} |
|
852 \ldots\\ |
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853 \isacommand{binder} bn \isacommand{where}\\ |
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854 \phantom{$|$} bn(ANil) $=$ $[]$\\ |
|
855 $|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)\\ |
|
856 \end{tabular}\bigskip |
|
857 |
|
858 \begin{center} |
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859 \mbox{\infer{\text{ANil} \approx_\alpha^{\text{bn}} \text{ANil}}{}}\bigskip |
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860 |
|
861 \mbox{\infer{\text{ACons}\;a\;t\;as \approx_\alpha^{\text{bn}} \text{ACons}\;a'\;t'\;as'} |
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862 {t \approx_\alpha t' & as \approx_\alpha^{bn} as'}} |
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863 \end{center} |
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864 |
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865 |
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866 \end{frame}} |
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867 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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868 *} |
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869 |
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870 |
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871 text_raw {* |
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872 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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873 \mode<presentation>{ |
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874 \begin{frame}<1> |
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875 \frametitle{\begin{tabular}{c}\LARGE``Raw'' Alpha-Equivalence\end{tabular}} |
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876 \mbox{}\\[6mm] |
|
877 |
|
878 \begin{center} |
|
879 LetRec as::assn t::trm \hspace{10mm}\isacommand{bind} bn(as) \isacommand{in} t \alert{as}\\ |
|
880 \end{center} |
|
881 |
|
882 |
|
883 \[\mbox{}\hspace{-4mm} |
|
884 \infer[\text{LetRec-}\!\approx_\alpha] |
|
885 {\text{LetRec}\;as\;t \approx_\alpha \text{LetRec}\;as'\;t'} |
|
886 {(\text{bn}(as), (t, as)) \approx\!\makebox[0mm][l]{${}_{\text{list}}$} |
|
887 ^{R,\text{fv}} (\text{bn}(as'), (t', as'))} |
|
888 \]\bigskip |
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889 |
|
890 \onslide<1->{\alert{deep recursive binders}} |
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891 \end{frame}} |
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892 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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893 *} |
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894 |
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895 text_raw {* |
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896 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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897 \mode<presentation>{ |
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898 \begin{frame}<1-> |
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899 \frametitle{\begin{tabular}{c}Restrictions\end{tabular}} |
|
900 \mbox{}\\[-6mm] |
|
901 |
|
902 Our restrictions on binding specifications: |
|
903 |
|
904 \begin{itemize} |
|
905 \item a body can only occur once in a list of binding clauses\medskip |
|
906 \item you can only have one binding function for a deep binder\medskip |
|
907 \item binding functions can return: the empty set, singletons, unions (similarly for lists) |
|
908 \end{itemize} |
|
909 |
|
910 |
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911 \end{frame}} |
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912 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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913 *} |
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914 |
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915 text_raw {* |
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916 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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917 \mode<presentation>{ |
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918 \begin{frame}<1-> |
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919 \frametitle{\begin{tabular}{c}Automatic Proofs\end{tabular}} |
|
920 \mbox{}\\[-6mm] |
|
921 |
|
922 \begin{itemize} |
|
923 \item we can show that $\alpha$'s are equivalence relations\medskip |
|
924 \item as a result we can use our quotient package to introduce the type(s) |
|
925 of $\alpha$-equated terms |
|
926 |
|
927 \[ |
|
928 \infer |
|
929 {\text{Lam}\;x\;t \alert{=} \text{Lam}\;x'\;t'} |
|
930 {\only<1>{([x], t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$} |
|
931 ^{=,\text{supp}} ([x'], t')}% |
|
932 \only<2>{[x].t = [x'].t'}} |
|
933 \] |
|
934 |
|
935 |
|
936 \item the properties for support are implied by the properties of $[\_].\_$ |
|
937 \item we can derive strong induction principles |
|
938 \end{itemize} |
|
939 |
|
940 |
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941 \end{frame}} |
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942 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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943 *} |
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944 |
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945 text_raw {* |
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946 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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947 \mode<presentation>{ |
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948 \begin{frame}<1>[t] |
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949 \frametitle{\begin{tabular}{c}Runtime is Acceptable\end{tabular}} |
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950 \mbox{}\\[-7mm]\mbox{} |
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951 |
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952 \footnotesize |
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953 \begin{center} |
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954 \begin{tikzpicture} |
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955 \draw (0,0) node[inner sep=2mm, ultra thick, draw=fg, rounded corners=2mm] |
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956 (A) {\begin{minipage}{0.8cm}bind.\\spec.\end{minipage}}; |
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957 |
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958 \draw (2,0) node[inner sep=2mm, ultra thick, draw=fg, rounded corners=2mm] |
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959 (B) {\begin{minipage}{0.8cm}raw\\terms\end{minipage}}; |
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960 |
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961 \draw (4,0) node[inner sep=2mm, ultra thick, draw=fg, rounded corners=2mm] |
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962 (C) {\begin{minipage}{0.8cm}$\alpha$-\\equiv.\end{minipage}}; |
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963 |
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964 \draw (0,-2) node[inner sep=2mm, ultra thick, draw=fg, rounded corners=2mm] |
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965 (D) {\begin{minipage}{0.8cm}quot.\\type\end{minipage}}; |
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966 |
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967 \draw (2,-2) node[inner sep=2mm, ultra thick, draw=fg, rounded corners=2mm] |
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968 (E) {\begin{minipage}{0.8cm}lift\\thms\end{minipage}}; |
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969 |
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970 \draw (4,-2) node[inner sep=2mm, ultra thick, draw=fg, rounded corners=2mm] |
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971 (F) {\begin{minipage}{0.8cm}add.\\thms\end{minipage}}; |
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972 |
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973 \draw[->,fg!50,line width=1mm] (A) -- (B); |
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974 \draw[->,fg!50,line width=1mm] (B) -- (C); |
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975 \draw[->,fg!50,line width=1mm, line join=round, rounded corners=2mm] |
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976 (C) -- (5,0) -- (5,-1) -- (-1,-1) -- (-1,-2) -- (D); |
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977 \draw[->,fg!50,line width=1mm] (D) -- (E); |
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978 \draw[->,fg!50,line width=1mm] (E) -- (F); |
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979 \end{tikzpicture} |
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980 \end{center} |
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981 |
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982 \begin{itemize} |
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983 \item Core Haskell: 11 types, 49 term-constructors, 7 binding functions |
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984 \begin{center} |
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985 $\sim$ 2 mins |
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986 \end{center} |
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987 \end{itemize} |
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988 |
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989 \end{frame}} |
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990 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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991 *} |
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992 |
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993 |
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994 text_raw {* |
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995 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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996 \mode<presentation>{ |
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997 \begin{frame}<1-> |
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998 \frametitle{\begin{tabular}{c}Interesting Phenomenon\end{tabular}} |
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999 \mbox{}\\[-6mm] |
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1000 |
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1001 \small |
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1002 \mbox{}\hspace{20mm} |
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1003 \begin{tabular}{ll} |
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1004 \multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\ |
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1005 \hspace{5mm}\phantom{$|$} Var name\\ |
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1006 \hspace{5mm}$|$ App trm trm\\ |
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1007 \hspace{5mm}$|$ Lam x::name t::trm |
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1008 & \isacommand{bind} x \isacommand{in} t\\ |
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1009 \hspace{5mm}$|$ Let as::assn t::trm |
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1010 & \isacommand{bind} bn(as) \isacommand{in} t\\ |
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1011 \multicolumn{2}{l}{\isacommand{and} assn $=$}\\ |
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1012 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\ |
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1013 \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\ |
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1014 \multicolumn{2}{l}{\isacommand{binder} bn \isacommand{where}}\\ |
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1015 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\ |
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1016 \multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\ |
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1017 \end{tabular}\bigskip\medskip |
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1018 |
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1019 we cannot quotient assn: ACons a \ldots $\not\approx_\alpha$ ACons b \ldots |
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1020 |
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1021 \only<1->{ |
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1022 \begin{textblock}{8}(0.2,7.3) |
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1023 \alert{\begin{tabular}{p{2.6cm}} |
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1024 \raggedright\footnotesize{}Should a ``naked'' assn be quotient? |
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1025 \end{tabular}\hspace{-3mm} |
|
1026 $\begin{cases} |
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1027 \mbox{} \\ \mbox{} |
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1028 \end{cases}$} |
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1029 \end{textblock}} |
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1030 \end{frame}} |
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1031 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1032 *} |
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1033 |
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1034 text_raw {* |
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1035 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1036 \mode<presentation>{ |
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1037 \begin{frame}<1-> |
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1038 \frametitle{\begin{tabular}{c}Conclusion\end{tabular}} |
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1039 \mbox{}\\[-6mm] |
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1040 |
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1041 \begin{itemize} |
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1042 \item the user does not see anything of the raw level\medskip |
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1043 \only<1>{\begin{center} |
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1044 Lam a (Var a) \alert{$=$} Lam b (Var b) |
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1045 \end{center}\bigskip} |
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1046 |
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1047 \item<2-> we have not yet done function definitions (will come soon and |
|
1048 we hope to make improvements over the old way there too)\medskip |
|
1049 \item<3-> it took quite some time to get here, but it seems worthwhile |
|
1050 (Barendregt's variable convention is unsound in general, |
|
1051 found bugs in two paper proofs, quotient package, POPL 2011 tutorial)\medskip |
|
1052 \end{itemize} |
|
1053 |
|
1054 |
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1055 \end{frame}} |
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1056 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1057 *} |
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1058 |
|
1059 text_raw {* |
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1060 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1061 \mode<presentation>{ |
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1062 \begin{frame}<1->[c] |
|
1063 \frametitle{\begin{tabular}{c}Future Work\end{tabular}} |
|
1064 \mbox{}\\[-6mm] |
|
1065 |
|
1066 \begin{itemize} |
|
1067 \item Function definitions |
|
1068 \end{itemize} |
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1069 |
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1070 \end{frame}} |
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1071 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1072 *} |
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1073 |
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1074 |
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1075 text_raw {* |
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1076 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1077 \mode<presentation>{ |
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1078 \begin{frame}<1->[c] |
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1079 \frametitle{\begin{tabular}{c}Questions?\end{tabular}} |
|
1080 \mbox{}\\[-6mm] |
|
1081 |
|
1082 \begin{center} |
|
1083 \alert{\huge{Thanks!}} |
|
1084 \end{center} |
|
1085 |
|
1086 \end{frame}} |
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1087 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1088 *} |
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1089 |
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1090 |
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1091 |
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1092 text_raw {* |
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1093 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1094 \mode<presentation>{ |
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1095 \begin{frame}<1-2>[c] |
|
1096 \frametitle{\begin{tabular}{c}Examples\end{tabular}} |
|
1097 \mbox{}\\[-6mm] |
|
1098 |
|
1099 \begin{center} |
|
1100 $(\{a,b\}, a \rightarrow b) \approx_\alpha (\{a, b\}, a \rightarrow b)$ |
|
1101 $(\{a,b\}, a \rightarrow b) \approx_\alpha (\{a, b\}, b \rightarrow a)$ |
|
1102 \end{center} |
|
1103 |
|
1104 \begin{center} |
|
1105 $(\{a,b\}, (a \rightarrow b, a \rightarrow b))$\\ |
|
1106 \hspace{17mm}$\not\approx_\alpha (\{a, b\}, (a \rightarrow b, b \rightarrow a))$ |
|
1107 \end{center} |
|
1108 |
|
1109 \onslide<2-> |
|
1110 {1.) \hspace{3mm}\isacommand{bind (set)} as \isacommand{in} $\tau_1$, |
|
1111 \isacommand{bind (set)} as \isacommand{in} $\tau_2$\medskip |
|
1112 |
|
1113 2.) \hspace{3mm}\isacommand{bind (set)} as \isacommand{in} $\tau_1$ $\tau_2$ |
|
1114 } |
|
1115 |
|
1116 \end{frame}} |
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1117 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1118 *} |
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1119 |
|
1120 (*<*) |
|
1121 end |
|
1122 (*>*) |
|