60 \end{center} |
74 \end{center} |
61 \end{frame}} |
75 \end{frame}} |
62 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
76 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
63 |
77 |
64 *} |
78 *} |
65 |
79 text_raw {* |
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80 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
67 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
81 \mode<presentation>{ |
68 \mode<presentation>{ |
82 \begin{frame}<1-4>[c] |
69 \begin{frame}<1-2> |
83 \frametitle{One Motivation} |
70 \frametitle{\begin{tabular}{c}Binding in Old Nominal\end{tabular}} |
84 |
71 \mbox{}\\[-6mm] |
85 \onslide<2->{Typing implemented in Prolog \textcolor{darkgray}{(from a textbook)}}\bigskip\\ |
72 |
86 |
73 \begin{itemize} |
87 \onslide<3->{ |
74 \item old Nominal provided a reasoning infrastructure for single binders\medskip |
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75 |
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76 \begin{center} |
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77 Lam [a].(Var a) |
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78 \end{center}\bigskip |
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79 |
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80 \item<2-> but representing |
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81 |
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82 \begin{center} |
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83 $\forall\{a_1,\ldots,a_n\}.\; T$ |
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84 \end{center}\medskip |
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85 |
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86 with single binders and reasoning about it is a \alert{\bf major} pain; |
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87 take my word for it! |
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88 \end{itemize} |
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89 |
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90 \only<1>{ |
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91 \begin{textblock}{6}(1.5,11) |
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92 \small |
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93 for example\\ |
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94 \begin{tabular}{l@ {\hspace{2mm}}l} |
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95 & a $\fresh$ Lam [a]. t\\ |
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96 & Lam [a]. (Var a) \alert{$=$} Lam [b]. (Var b)\\ |
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97 & Barendregt style reasoning about bound variables\\ |
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98 \end{tabular} |
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99 \end{textblock}} |
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100 |
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101 \end{frame}} |
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102 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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103 *} |
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104 |
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105 text_raw {* |
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106 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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107 \mode<presentation>{ |
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108 \begin{frame}<1-4> |
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109 \frametitle{\begin{tabular}{c}Binding Sets of Names\end{tabular}} |
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110 \mbox{}\\[-3mm] |
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111 |
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112 \begin{itemize} |
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113 \item binding sets of names has some interesting properties:\medskip |
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114 |
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115 \begin{center} |
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116 \begin{tabular}{l} |
88 \begin{tabular}{l} |
117 $\forall\{x, y\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{y, x\}.\, y \rightarrow x$ |
89 type (Gamma, var(X), T) :- member (X,T) Gamma.\smallskip\medskip\\ |
118 \bigskip\smallskip\\ |
90 |
119 |
91 type (Gamma, app(M, N), T') :-\\ |
120 \onslide<2->{% |
92 \hspace{3cm}type (Gamma, M, arrow(T, T')),\\ |
121 $\forall\{x, y\}.\, x \rightarrow y \;\;\not\approx_\alpha\;\; \forall\{z\}.\, z \rightarrow z$ |
93 \hspace{3cm}type (Gamma, N, T).\smallskip\medskip\\ |
122 }\bigskip\smallskip\\ |
94 |
123 |
95 type (Gamma, lam(X, M), arrow(T, T')) :-\\ |
124 \onslide<3->{% |
96 \hspace{3cm}type ((X, T)::Gamma, M, T').\smallskip\medskip\\ |
125 $\forall\{x\}.\, x \rightarrow y \;\;\approx_\alpha\;\; \forall\{x, \alert{z}\}.\, x \rightarrow y$ |
97 |
126 }\medskip\\ |
98 member X X::Tail.\\ |
127 \onslide<3->{\hspace{4cm}\small provided $z$ is fresh for the type} |
99 member X Y::Tail :- member X Tail.\\ |
128 \end{tabular} |
100 \end{tabular}} |
129 \end{center} |
101 |
130 \end{itemize} |
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131 |
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132 \begin{textblock}{8}(2,14.5) |
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133 \footnotesize $^*$ $x$, $y$, $z$ are assumed to be distinct |
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134 \end{textblock} |
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135 |
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136 \only<4>{ |
102 \only<4>{ |
137 \begin{textblock}{6}(2.5,4) |
103 \begin{textblock}{6}(2.5,2) |
138 \begin{tikzpicture} |
104 \begin{tikzpicture} |
139 \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
105 \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
140 {\normalsize\color{darkgray} |
106 {\color{darkgray} |
141 \begin{minipage}{8cm}\raggedright |
107 \begin{minipage}{8cm}\raggedright |
142 For type-schemes the order of bound names does not matter, and |
108 The problem is that \smath{\lambda x.\lambda x. (x\;x)} |
143 alpha-equivalence is preserved under \alert{vacuous} binders. |
109 gets the types |
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110 \begin{center} |
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111 \begin{tabular}{l} |
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112 \smath{T\rightarrow (T\rightarrow S) \rightarrow S} and\\ |
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113 \smath{(T\rightarrow S)\rightarrow T \rightarrow S}\\ |
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114 \end{tabular} |
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115 \end{center} |
144 \end{minipage}}; |
116 \end{minipage}}; |
145 \end{tikzpicture} |
117 \end{tikzpicture} |
146 \end{textblock}} |
118 \end{textblock}} |
147 \end{frame}} |
119 |
148 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
120 \end{frame}} |
149 *} |
121 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
150 |
122 *} |
151 text_raw {* |
123 |
152 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
124 text_raw {* |
153 \mode<presentation>{ |
125 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
154 \begin{frame}<1-3> |
126 \mode<presentation>{ |
155 \frametitle{\begin{tabular}{c}Other Binding Modes\end{tabular}} |
127 \begin{frame}<1>[c] |
156 \mbox{}\\[-3mm] |
128 \frametitle{Higher-Order Unification} |
157 |
129 |
158 \begin{itemize} |
130 \begin{itemize} |
159 \item alpha-equivalence being preserved under vacuous binders is \underline{not} always |
131 \item Lambda Prolog with full Higher-Order Unification\\ |
160 wanted:\bigskip\bigskip\normalsize |
132 \textcolor{darkgray}{(no mgus, undecidable, modulo $\alpha\beta$)}\bigskip |
161 |
133 \item Higher-Order Pattern Unification\\ |
162 \begin{tabular}{@ {\hspace{-8mm}}l} |
134 \textcolor{darkgray}{(has mgus, decidable, some restrictions, modulo $\alpha\beta_0$)} |
163 $\text{let}\;x = 3\;\text{and}\;y = 2\;\text{in}\;x - y\;\text{end}$\medskip\\ |
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164 \onslide<2->{$\;\;\;\only<2>{\approx_\alpha}\only<3>{\alert{\not\approx_\alpha}} |
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165 \text{let}\;y = 2\;\text{and}\;x = 3\only<3->{\alert{\;\text{and} |
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166 \;z = \text{loop}}}\;\text{in}\;x - y\;\text{end}$} |
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167 \end{tabular} |
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168 |
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169 |
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170 \end{itemize} |
135 \end{itemize} |
171 |
136 |
172 \end{frame}} |
137 \end{frame}} |
173 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
138 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
174 *} |
139 *} |
175 |
140 |
176 text_raw {* |
141 text_raw {* |
177 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
142 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
178 \mode<presentation>{ |
143 \mode<presentation>{ |
179 \begin{frame}<1> |
144 \begin{frame}<1-10>[t] |
180 \frametitle{\begin{tabular}{c}\LARGE{}Even Another Binding Mode\end{tabular}} |
145 \frametitle{Underlying Ideas} |
181 \mbox{}\\[-3mm] |
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182 |
146 |
183 \begin{itemize} |
147 \begin{itemize} |
184 \item sometimes one wants to abstract more than one name, but the order \underline{does} matter\bigskip |
148 \item<1-> Unification (\alert{only}) up to $\alpha$ |
185 |
149 |
186 \begin{center} |
150 \item<2-> Swappings / Permutations |
187 \begin{tabular}{@ {\hspace{-8mm}}l} |
151 |
188 $\text{let}\;(x, y) = (3, 2)\;\text{in}\;x - y\;\text{end}$\medskip\\ |
152 \only<2-5>{ |
189 $\;\;\;\not\approx_\alpha |
153 \begin{center} |
190 \text{let}\;(y, x) = (3, 2)\;\text{in}\;x - y\;\text{end}$ |
154 \begin{tabular}{r@ {\hspace{1mm}}l@ {\hspace{12mm}}r@ {\hspace{1mm}}l} |
191 \end{tabular} |
155 \only<2>{\smath{\textcolor{white}{[b\!:=\!a]}}}% |
192 \end{center} |
156 \only<3>{\smath{[b\!:=\!a]}}% |
193 |
157 \only<4-5>{\smath{\alert{\swap{a}{b}\,\act}}} & |
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158 \onslide<2-5>{\smath{\lambda a.b}} & |
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159 |
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160 \only<2>{\smath{\textcolor{white}{[b\!:=\!a]}}}% |
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161 \only<3>{\smath{[b\!:=\!a]}}% |
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162 \only<4-5>{\smath{\alert{\swap{a}{b}\,\act}}} & |
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163 \onslide<2-5>{\smath{\lambda c.b}}\\ |
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164 |
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165 \onslide<3-5>{\smath{=}} & \only<3>{\smath{\lambda a.a}}\only<4-5>{\smath{\lambda b.a}} & |
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166 \onslide<3-5>{\smath{=}} & \only<3>{\smath{\lambda c.a}}\only<4-5>{\smath{\lambda c.a}}\\ |
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167 \end{tabular} |
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168 \end{center}\bigskip |
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169 |
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170 \onslide<4-5>{ |
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171 \begin{center} |
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172 \begin{tikzpicture} |
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173 \draw (0,0) node[inner sep=0mm,fill=cream, ultra thick, draw=cream] |
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174 {\begin{minipage}{8cm} |
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175 \begin{tabular}{r@ {\hspace{3mm}}l} |
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176 \smath{\swap{a}{b}\act t} $\;\dn$ & \alert{swap} {\bf all} occurences of\\ |
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177 & \smath{b} and \smath{a} in \smath{t} |
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178 \end{tabular} |
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179 \end{minipage}}; |
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180 \end{tikzpicture} |
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181 \end{center}}\bigskip |
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182 |
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183 \onslide<5>{ |
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184 Unlike for \smath{[b\!:=\!a]\act(-)}, for \smath{\swap{a}{b}\act (-)} we do |
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185 have if \smath{t =_\alpha t'} then \smath{\pi \act t =_\alpha \pi \act t'.}}} |
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186 |
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187 \item<6-> Variables (or holes)\bigskip |
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188 |
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189 \begin{center} |
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190 \onslide<7->{\mbox{}\hspace{-25mm}\smath{\lambda x\hspace{-0.5mm}s .}} |
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191 \onslide<8-9>{\raisebox{-1.7mm}{\huge\smath{(}}}\raisebox{-4mm}{\begin{tikzpicture} |
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192 \fill[blue] (0, 0) circle (5mm); |
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193 \end{tikzpicture}} |
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194 \onslide<8-9>{\smath{y\hspace{-0.5mm}s}{\raisebox{-1.7mm}{\huge\smath{)}}}}\bigskip |
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195 \end{center} |
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196 |
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197 \only<8-9>{\smath{y\hspace{-0.5mm}s} are the parameters the hole can depend on\onslide<9->{, but |
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198 then you need $\beta_0$-reduction\medskip |
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199 \begin{center} |
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200 \smath{(\lambda x. t) y \longrightarrow_{\beta_0} t[x:=y]} |
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201 \end{center}}} |
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202 |
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203 \only<10>{we will record the information about which parameters a hole |
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204 \alert{\bf cannot} depend on} |
194 |
205 |
195 \end{itemize} |
206 \end{itemize} |
196 |
207 |
197 \end{frame}} |
208 \end{frame}} |
198 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
209 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
199 *} |
210 *} |
200 |
211 |
201 text_raw {* |
212 text_raw {* |
202 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
213 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
203 \mode<presentation>{ |
214 \mode<presentation>{ |
204 \begin{frame}<1-2> |
215 \begin{frame}<1-4>[c] |
205 \frametitle{\begin{tabular}{c}\LARGE{}Three Binding Modes\end{tabular}} |
216 \frametitle{Terms} |
206 \mbox{}\\[-3mm] |
217 |
207 |
218 \begin{tabular}{lll @ {\hspace{10mm}}lll} |
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219 |
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220 \onslide<1->{\pgfuseshading{smallbluesphere}} & |
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221 \onslide<1->{\colorbox{cream}{\smath{\unit}}} & |
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222 \onslide<1->{Units} & |
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223 |
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224 \onslide<2->{\pgfuseshading{smallbluesphere}} & |
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225 \onslide<2->{\colorbox{cream}{\smath{a}}} & |
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226 \onslide<2->{Atoms} \\[5mm] |
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227 |
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228 \onslide<1->{\pgfuseshading{smallbluesphere}} & |
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229 \onslide<1->{\colorbox{cream}{\smath{\pair{t}{t'}}}} & |
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230 \onslide<1->{Pairs} & |
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231 |
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232 \onslide<3->{\pgfuseshading{smallbluesphere}} & |
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233 \onslide<3->{\colorbox{cream}{\smath{\abst{a}{t}}}} & |
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234 \onslide<3->{Abstractions}\\[5mm] |
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235 |
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236 \onslide<1->{\pgfuseshading{smallbluesphere}} & |
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237 \onslide<1->{\colorbox{cream}{\smath{\app{F}{t}}}} & |
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238 \onslide<1->{Funct.} & |
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239 |
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240 \onslide<4->{\pgfuseshading{smallbluesphere}} & |
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241 \onslide<4->{\colorbox{cream}{\smath{\pi\susp X}}} & |
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242 \onslide<4->{Suspensions} |
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243 \end{tabular} |
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244 |
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245 \only<2>{ |
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246 \begin{textblock}{13}(1.5,12) |
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247 \small Atoms are constants \textcolor{darkgray}{(infinitely many of them)} |
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248 \end{textblock}} |
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249 |
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250 \only<3>{ |
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251 \begin{textblock}{13}(1.5,12) |
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252 \small \smath{\ulcorner \lambda\abst{a}{a}\urcorner \mapsto \text{fn\ }\abst{a}{a}}\\ |
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253 \small constructions like \smath{\text{fn\ }\abst{X}{X}} are not allowed |
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254 \end{textblock}} |
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255 |
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256 \only<4>{ |
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257 \begin{textblock}{13}(1.5,12) |
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258 \small \smath{X} is a variable standing for a term\\ |
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259 \small \smath{\pi} is an explicit permutation \smath{\swap{a_1}{b_1}\ldots\swap{a_n}{b_n}}, |
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260 waiting to be applied to the term that is substituted for \smath{X} |
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261 \end{textblock}} |
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262 |
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263 \end{frame}} |
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264 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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265 *} |
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266 |
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267 text_raw {* |
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268 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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269 \mode<presentation>{ |
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270 \begin{frame}<1-3>[c] |
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271 \frametitle{Permutations} |
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272 |
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273 a permutation applied to a term |
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274 |
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275 \begin{center} |
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276 \begin{tabular}{lrcl} |
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277 \pgfuseshading{smallbluesphere} & |
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278 \smath{[]\act c} & \smath{\dn} & \smath{c} \\ |
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279 |
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280 \pgfuseshading{smallbluesphere} & |
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281 \smath{\swap{a}{b}\!::\!\pi\act c} & \smath{\dn} & |
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282 \smath{\begin{cases} |
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283 a & \text{if}\;\pi\act c = b\\ |
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284 b & \text{if}\;\pi\act c = a\\ |
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285 \pi\act c & \text{otherwise} |
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286 \end{cases}}\\ |
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287 |
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288 \onslide<2->{\pgfuseshading{smallbluesphere}} & |
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289 \onslide<2->{\smath{\pi\act\abst{a}{t}}} & \onslide<2->{\smath{\dn}} & |
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290 \onslide<2->{\smath{\abst{\pi\act a}{\pi\act t}}}\\ |
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291 |
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292 \onslide<3->{\pgfuseshading{smallbluesphere}} & |
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293 \onslide<3->{\smath{\pi\act\pi'\act X}} & \onslide<3->{\smath{\dn}} & |
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294 \onslide<3->{\smath{(\pi @ \pi')\act X}}\\ |
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295 \end{tabular} |
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296 \end{center} |
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297 |
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298 \end{frame}} |
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299 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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300 *} |
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301 |
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302 text_raw {* |
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303 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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304 \mode<presentation>{ |
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305 \begin{frame}<1-3>[c] |
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306 \frametitle{Freshness Constraints} |
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307 |
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308 Recall \smath{\lambda a. \raisebox{-0.7mm}{\tikz \fill[blue] (0, 0) circle (2.5mm);}} |
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309 \bigskip\pause |
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310 |
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311 We therefore will identify |
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312 |
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313 \begin{center} |
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314 \smath{\mathtt{fn\ } a. X \;\approx\; \mathtt{fn\ } b. \alert<3->{\swap{a}{b}}\act X} |
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315 \end{center} |
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316 |
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317 provided that `\smath{b} is fresh for \smath{X} --- (\smath{b\fresh X})', |
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318 i.e., does not occur freely in any ground term that might be substituted for |
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319 \smath{X}.\bigskip\pause |
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320 |
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321 If we know more about \smath{X}, e.g., if we knew that \smath{a\fresh X} and |
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322 \smath{b\fresh X}, then we can replace\\ \smath{\swap{a}{b}\act X} by |
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323 \smath{X}. |
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324 |
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325 \end{frame}} |
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326 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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327 *} |
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328 |
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329 text_raw {* |
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330 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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331 \mode<presentation>{ |
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332 \begin{frame}<1-4>[c] |
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333 \frametitle{Equivalence Judgements} |
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334 |
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335 \alt<1>{Our equality is {\bf not} just}{but judgements} |
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336 |
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337 \begin{center} |
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338 \begin{tabular}{rl} |
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339 \colorbox{cream}{\smath{\onslide<2->{\nabla \vdash} t \approx t'}} & \alert{$\alpha$-equivalence}\\[1mm] |
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340 \onslide<4->{\colorbox{cream}{\smath{\onslide<2->{\nabla \vdash} a \fresh t}}} & |
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341 \onslide<4->{\alert{freshness}} |
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342 \end{tabular} |
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343 \end{center} |
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344 |
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345 \onslide<2->{ |
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346 where |
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347 \begin{center} |
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348 \smath{\nabla = \{a_1\fresh X_1,\ldots, a_n\fresh X_n\}} |
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349 \end{center} |
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350 is a finite set of \alert{freshness assumptions}.} |
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351 |
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352 \onslide<3->{ |
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353 \begin{center} |
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354 \smath{\{a\fresh X,b\fresh X\} \vdash \text{fn\ } a. X \approx \text{fn\ } b. X} |
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355 \end{center}} |
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356 |
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357 \end{frame}} |
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358 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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359 *} |
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360 |
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361 text_raw {* |
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362 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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363 \mode<presentation>{ |
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364 \begin{frame}<1>[c] |
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365 \frametitle{Rules for Equivalence} |
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366 |
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367 \begin{center} |
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368 \begin{tabular}{c} |
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369 Excerpt\\ |
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370 (i.e.~only the interesting rules) |
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371 \end{tabular} |
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372 \end{center} |
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373 |
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374 \end{frame}} |
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375 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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376 *} |
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377 |
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378 text_raw {* |
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379 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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380 \mode<presentation>{ |
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381 \begin{frame}<1>[c] |
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382 \frametitle{Rules for Equivalence} |
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383 |
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384 \begin{center} |
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385 \begin{tabular}{c} |
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386 \colorbox{cream}{\smath{\infer{\nabla \vdash a \approx a}{}}}\\[8mm] |
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387 |
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388 \colorbox{cream}{% |
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389 \smath{\infer{\nabla \vdash \abst{a}{t} \approx \abst{a}{t'}} |
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390 {\nabla \vdash t \approx t'}}}\\[8mm] |
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391 |
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392 \colorbox{cream}{% |
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393 \smath{\infer{\nabla \vdash \abst{a}{t} \approx \abst{b}{t'}} |
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394 {a\not=b\;\; & \nabla \vdash t \approx \swap{a}{b}\act t'\;\;& \nabla \vdash a\fresh t'}}} |
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395 \end{tabular} |
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396 \end{center} |
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397 |
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398 \end{frame}} |
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399 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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400 *} |
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401 |
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402 text_raw {* |
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403 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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404 \mode<presentation>{ |
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405 \begin{frame}<1-3>[c] |
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406 \frametitle{Rules for Equivalence} |
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407 |
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408 \begin{center} |
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409 \colorbox{cream}{% |
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410 \smath{% |
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411 \infer{\nabla \vdash \pi\act X \approx \pi'\act X} |
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412 {\begin{array}{c} |
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413 (a\fresh X)\in\nabla\\ |
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414 \text{for all}\; a \;\text{with}\;\pi\act a \not= \pi'\act a |
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415 \end{array} |
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416 }}} |
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417 \end{center} |
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418 |
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419 \onslide<2->{ |
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420 for example\\[4mm] |
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421 |
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422 \alt<2>{% |
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423 \begin{center} |
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424 \smath{\{a\fresh\!X, b\fresh\!X\} \vdash X \approx \swap{a}{b}\act X} |
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425 \end{center}} |
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426 {% |
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427 \begin{center} |
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428 \smath{\{a\fresh\!X, c\fresh\!X\} \vdash \swap{a}{c}\swap{a}{b}\act X \approx \swap{b}{c}\act X} |
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429 \end{center}} |
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430 |
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431 \onslide<3->{ |
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432 \begin{tabular}{@ {}lllll@ {}} |
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433 because & |
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434 \smath{\swap{a}{c}\swap{a}{b}}: & |
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435 \smath{a\mapsto b} & |
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436 \smath{\swap{b}{c}}: & |
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437 \smath{a\mapsto a}\\ |
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438 & & \smath{b\mapsto c} & & \smath{b\mapsto c}\\ |
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439 & & \smath{c\mapsto a} & & \smath{c\mapsto b}\\ |
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440 \end{tabular} |
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441 disagree at \smath{a} and \smath{c}.} |
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442 } |
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443 |
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444 \end{frame}} |
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445 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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446 *} |
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447 |
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448 text_raw {* |
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449 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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450 \mode<presentation>{ |
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451 \begin{frame}<1>[c] |
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452 \frametitle{Rules for Freshness} |
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453 |
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454 \begin{center} |
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455 \begin{tabular}{c} |
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456 Excerpt\\ |
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457 (i.e.~only the interesting rules) |
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458 \end{tabular} |
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459 \end{center} |
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460 |
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461 \end{frame}} |
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462 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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463 *} |
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464 |
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465 text_raw {* |
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466 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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467 \mode<presentation>{ |
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468 \begin{frame}<1>[c] |
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469 \frametitle{Rules for Freshness} |
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470 |
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471 \begin{center} |
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472 \begin{tabular}{c} |
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473 \colorbox{cream}{% |
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474 \smath{\infer{\nabla \vdash a\fresh b}{a\not= b}}}\\[5mm] |
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475 |
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476 \colorbox{cream}{% |
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477 \smath{\infer{\nabla \vdash a\fresh\abst{a}{t}}{}}}\hspace{7mm} |
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478 \colorbox{cream}{% |
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479 \smath{\infer{\nabla \vdash a\fresh\abst{b}{t}} |
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480 {a\not= b\;\; & \nabla \vdash a\fresh t}}}\\[5mm] |
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481 |
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482 \colorbox{cream}{% |
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483 \smath{\infer{\nabla \vdash a\fresh \pi\act X} |
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484 {(\pi^{-1}\act a\fresh X)\in\nabla}}} |
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485 \end{tabular} |
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486 \end{center} |
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487 |
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488 \end{frame}} |
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489 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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490 *} |
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491 |
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492 text_raw {* |
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493 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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494 \mode<presentation>{ |
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495 \begin{frame}<1-4>[t] |
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496 \frametitle{$\approx$ is an Equivalence} |
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497 \mbox{}\\[5mm] |
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498 |
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499 \begin{center} |
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500 \colorbox{cream}{\alert{Theorem:} |
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501 $\approx$ is an equivalence relation.} |
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502 \end{center}\bigskip |
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503 |
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504 \only<1>{% |
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505 \begin{tabular}{ll} |
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506 (Reflexivity) & $\smath{\nabla\vdash t\approx t}$\\[2mm] |
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507 (Symmetry) & if $\smath{\nabla\vdash t_1\approx t_2}\;$ |
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508 then $\;\smath{\nabla\vdash t_2\approx t_1}$\\[2mm] |
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509 (Transitivity) & if $\smath{\nabla\vdash t_1\approx t_2}\;$ and |
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510 $\;\smath{\nabla\vdash t_2\approx t_3}$\\ |
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511 & then $\smath{\nabla\vdash t_1\approx t_3}$\\ |
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512 \end{tabular}} |
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513 |
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514 \only<2->{% |
208 \begin{itemize} |
515 \begin{itemize} |
209 \item the order does not matter and alpha-equivelence is preserved under |
516 \item<2-> \smath{\nabla \vdash t\approx t'} then \smath{\nabla \vdash \pi\act t\approx \pi\act t'} |
210 vacuous binders \textcolor{gray}{(restriction)}\medskip |
517 |
211 |
518 \item<2-> \smath{\nabla \vdash a\fresh t} then |
212 \item the order does not matter, but the cardinality of the binders |
519 \smath{\nabla \vdash \pi\act a\fresh \pi\act t} |
213 must be the same \textcolor{gray}{(abstraction)}\medskip |
520 |
214 |
521 \item<3-> \smath{\nabla \vdash t\approx \pi\act t'} then |
215 \item the order does matter |
522 \smath{\nabla \vdash (\pi^{-1})\act t\approx t'} |
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523 |
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524 \item<3-> \smath{\nabla \vdash a\fresh \pi\act t} then |
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525 \smath{\nabla \vdash (\pi^{-1})\act a\fresh t} |
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526 |
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527 \item<4-> \smath{\nabla \vdash a\fresh t} and \smath{\nabla \vdash t\approx t'} then |
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528 \smath{\nabla \vdash a\fresh t'} |
216 \end{itemize} |
529 \end{itemize} |
217 |
530 } |
218 \onslide<2->{ |
531 |
219 \begin{center} |
532 \end{frame}} |
220 \isacommand{bind\_res}\hspace{6mm} |
533 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
221 \isacommand{bind\_set}\hspace{6mm} |
534 *} |
222 \isacommand{bind} |
535 |
223 \end{center}} |
536 text_raw {* |
224 |
537 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
225 \end{frame}} |
538 \mode<presentation>{ |
226 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
539 \begin{frame}<1-4> |
227 *} |
540 \frametitle{Comparison $=_\alpha$} |
228 |
541 |
229 text_raw {* |
542 Traditionally \smath{=_\alpha} is defined as |
230 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
543 |
231 \mode<presentation>{ |
544 \begin{center} |
232 \begin{frame}<1-3> |
545 \colorbox{cream}{% |
233 \frametitle{\begin{tabular}{c}Specification of Binding\end{tabular}} |
546 \begin{minipage}{9cm} |
234 \mbox{}\\[-6mm] |
547 \raggedright least congruence which identifies \smath{\abst{a}{t}} |
235 |
548 with \smath{\abst{b}{[a:=b]t}} provided \smath{b} is not free |
236 \mbox{}\hspace{10mm} |
549 in \smath{t} |
237 \begin{tabular}{ll} |
550 \end{minipage}} |
238 \multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\ |
551 \end{center} |
239 \hspace{5mm}\phantom{$|$} Var name\\ |
552 |
240 \hspace{5mm}$|$ App trm trm\\ |
553 where \smath{[a:=b]t} replaces all free occurrences of\\ |
241 \hspace{5mm}$|$ Lam \only<2->{x::}name \only<2->{t::}trm |
554 \smath{a} by \smath{b} in \smath{t}. |
242 & \onslide<2->{\isacommand{bind} x \isacommand{in} t}\\ |
555 \bigskip |
243 \hspace{5mm}$|$ Let \only<2->{as::}assn \only<2->{t::}trm |
556 |
244 & \onslide<2->{\isacommand{bind} bn(as) \isacommand{in} t}\\ |
557 \only<2>{% |
245 \multicolumn{2}{l}{\isacommand{and} assn $=$}\\ |
558 \begin{textblock}{13}(1.2,10) |
246 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\ |
559 For \alert{ground} terms: |
247 \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\ |
560 |
248 \multicolumn{2}{l}{\onslide<3->{\isacommand{binder} bn \isacommand{where}}}\\ |
561 \begin{center} |
249 \multicolumn{2}{l}{\onslide<3->{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ []}}\\ |
562 \colorbox{cream}{% |
250 \multicolumn{2}{l}{\onslide<3->{\hspace{5mm}$|$ bn(ACons a t as) $=$ [a] @ bn(as)}}\\ |
563 \begin{minipage}{9.0cm} |
251 \end{tabular} |
564 \begin{tabular}{@ {}rl} |
252 |
565 \underline{Theorem:} |
253 |
566 & \smath{t=_\alpha t'\;\;} if\hspace{-0.5mm}f~\smath{\;\;\emptyset \vdash t\approx t'}\\[2mm] |
254 |
567 & \smath{a\not\in FA(t)\;\;} if\hspace{-0.5mm}f~\smath{\;\;\emptyset\vdash a\fresh t} |
255 \end{frame}} |
568 \end{tabular} |
256 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
569 \end{minipage}} |
257 *} |
570 \end{center} |
258 |
571 \end{textblock}} |
259 text_raw {* |
572 |
260 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
573 \only<3>{% |
261 \mode<presentation>{ |
574 \begin{textblock}{13}(1.2,10) |
262 \begin{frame}<1-5> |
575 In general \smath{=_\alpha} and \smath{\approx} are distinct! |
263 \frametitle{\begin{tabular}{c}Inspiration from Ott\end{tabular}} |
576 \begin{center} |
264 \mbox{}\\[-3mm] |
577 \colorbox{cream}{% |
265 |
578 \begin{minipage}{6.0cm} |
266 \begin{itemize} |
579 \smath{\abst{a}{X}=_\alpha \abst{b}{X}\;} but not\\[2mm] |
267 \item this way of specifying binding is inspired by |
580 \smath{\emptyset \vdash \abst{a}{X} \approx \abst{b}{X}\;} (\smath{a\not=b}) |
268 Ott\onslide<2->{, \alert{\bf but} we made adjustments:}\medskip |
581 \end{minipage}} |
269 |
582 \end{center} |
270 |
583 \end{textblock}} |
271 \only<2>{ |
584 |
272 \begin{itemize} |
585 \only<4>{ |
273 \item Ott allows specifications like\smallskip |
586 \begin{textblock}{6}(1,2) |
274 \begin{center} |
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275 $t ::= t\;t\; |\;\lambda x.t$ |
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276 \end{center} |
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277 \end{itemize}} |
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278 |
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279 \only<3-4>{ |
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280 \begin{itemize} |
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281 \item whether something is bound can depend in Ott on other bound things\smallskip |
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282 \begin{center} |
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283 \begin{tikzpicture} |
587 \begin{tikzpicture} |
284 \node (A) at (-0.5,1) {Foo $(\lambda y. \lambda x. t)$}; |
588 \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
285 \node (B) at ( 1.1,1) {$s$}; |
589 {\color{darkgray} |
286 \onslide<4>{\node (C) at (0.5,0) {$\{y, x\}$};} |
590 \begin{minipage}{10cm}\raggedright |
287 \onslide<4>{\draw[->,red,line width=1mm] (A) -- (C);} |
591 That is a crucial point: if we had\\[-2mm] |
288 \onslide<4>{\draw[->,red,line width=1mm] (C) -- (B);} |
592 \[\smath{\emptyset \vdash \abst{a}{X}\approx \abst{b}{X}}\mbox{,}\] |
289 \end{tikzpicture} |
593 then applying $\smath{[X:=a]}$, $\smath{[X:=b]}$, $\ldots$\\ |
290 \end{center} |
594 give two terms that are {\bf not} $\alpha$-equivalent.\\[3mm] |
291 \onslide<4>{this might make sense for ``raw'' terms, but not at all |
595 The freshness constraints $\smath{a\fresh X}$ and $\smath{b\fresh X}$ |
292 for $\alpha$-equated terms} |
596 rule out the problematic substitutions. Therefore |
293 \end{itemize}} |
597 |
294 |
598 \[\smath{\{a\fresh X,b\fresh X\} \vdash \abst{a}{X}\approx \abst{b}{X}}\] |
295 \only<5>{ |
599 |
296 \begin{itemize} |
600 does hold. |
297 \item we allow multiple binders and bodies\smallskip |
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298 \begin{center} |
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299 \isacommand{bind} a b c \isacommand{in} x y z |
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300 \end{center}\bigskip\medskip |
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301 the reason is that in general |
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302 \begin{center} |
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303 \begin{tabular}{rcl} |
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304 \isacommand{bind\_res} as \isacommand{in} x y & $\not\Leftrightarrow$ & |
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305 \begin{tabular}{@ {}l} |
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306 \isacommand{bind\_res} as \isacommand{in} x,\\ |
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307 \isacommand{bind\_res} as \isacommand{in} y |
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308 \end{tabular} |
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309 \end{tabular} |
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310 \end{center}\smallskip |
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311 |
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312 same with \isacommand{bind\_set} |
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313 \end{itemize}} |
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314 \end{itemize} |
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315 |
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316 |
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317 \end{frame}} |
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318 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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319 *} |
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320 |
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321 text_raw {* |
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322 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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323 \mode<presentation>{ |
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324 \begin{frame}<1> |
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325 \frametitle{\begin{tabular}{c}Alpha-Equivalence\end{tabular}} |
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326 \mbox{}\\[-3mm] |
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327 |
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328 \begin{itemize} |
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329 \item in old Nominal, we represented single binders as partial functions:\bigskip |
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330 |
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331 \begin{center} |
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332 \begin{tabular}{l} |
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333 Lam [$a$].\,$t$ $\;{^\text{``}}\!\dn{}\!^{\text{''}}$\\[2mm] |
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334 \;\;\;\;$\lambda b.$\;$\text{if}\;a = b\;\text{then}\;t\;\text{else}$\\ |
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335 \phantom{\;\;\;\;$\lambda b.$\;\;\;\;\;\;}$\text{if}\;b \fresh t\; |
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336 \text{then}\;(a\;b)\act t\;\text{else}\;\text{error}$ |
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337 \end{tabular} |
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338 \end{center} |
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339 \end{itemize} |
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340 |
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341 \begin{textblock}{10}(2,14) |
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342 \footnotesize $^*$ alpha-equality coincides with equality on functions |
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343 \end{textblock} |
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344 \end{frame}} |
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345 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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346 *} |
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347 |
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348 text_raw {* |
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349 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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350 \mode<presentation>{ |
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351 \begin{frame}<1-> |
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352 \frametitle{\begin{tabular}{c}New Design\end{tabular}} |
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353 \mbox{}\\[4mm] |
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354 |
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355 \begin{center} |
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356 \begin{tikzpicture} |
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357 \alt<2> |
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358 {\draw (0,0) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm] |
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359 (A) {\textcolor{red}{\begin{minipage}{1.1cm}bind.\\spec.\end{minipage}}};} |
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360 {\draw (0,0) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm] |
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361 (A) {\begin{minipage}{1.1cm}bind.\\spec.\end{minipage}};} |
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362 |
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363 \alt<3> |
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364 {\draw (3,0) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm] |
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365 (B) {\textcolor{red}{\begin{minipage}{1.1cm}raw\\terms\end{minipage}}};} |
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366 {\draw (3,0) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm] |
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367 (B) {\begin{minipage}{1.1cm}raw\\terms\end{minipage}};} |
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368 |
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369 \alt<4> |
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370 {\draw (6,0) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm] |
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371 (C) {\textcolor{red}{\begin{minipage}{1.1cm}$\alpha$-\\equiv.\end{minipage}}};} |
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372 {\draw (6,0) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm] |
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373 (C) {\begin{minipage}{1.1cm}$\alpha$-\\equiv.\end{minipage}};} |
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374 |
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375 \alt<5> |
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376 {\draw (0,-3) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm] |
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377 (D) {\textcolor{red}{\begin{minipage}{1.1cm}quot.\\type\end{minipage}}};} |
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378 {\draw (0,-3) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm] |
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379 (D) {\begin{minipage}{1.1cm}quot.\\type\end{minipage}};} |
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380 |
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381 \alt<6> |
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382 {\draw (3,-3) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm] |
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383 (E) {\textcolor{red}{\begin{minipage}{1.1cm}lift\\thms\end{minipage}}};} |
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384 {\draw (3,-3) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm] |
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385 (E) {\begin{minipage}{1.1cm}lift\\thms\end{minipage}};} |
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386 |
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387 \alt<7> |
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388 {\draw (6,-3) node[inner sep=3mm, ultra thick, draw=red, rounded corners=2mm] |
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389 (F) {\textcolor{red}{\begin{minipage}{1.1cm}add.\\thms\end{minipage}}};} |
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390 {\draw (6,-3) node[inner sep=3mm, ultra thick, draw=white, rounded corners=2mm] |
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391 (F) {\begin{minipage}{1.1cm}add.\\thms\end{minipage}};} |
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392 |
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393 \draw[->,white!50,line width=1mm] (A) -- (B); |
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394 \draw[->,white!50,line width=1mm] (B) -- (C); |
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395 \draw[->,white!50,line width=1mm, line join=round, rounded corners=2mm] |
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396 (C) -- (8,0) -- (8,-1.5) -- (-2,-1.5) -- (-2,-3) -- (D); |
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397 \draw[->,white!50,line width=1mm] (D) -- (E); |
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398 \draw[->,white!50,line width=1mm] (E) -- (F); |
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399 \end{tikzpicture} |
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400 \end{center} |
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401 |
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402 \end{frame}} |
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403 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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404 *} |
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405 |
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406 |
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407 |
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408 text_raw {* |
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409 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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410 \mode<presentation>{ |
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411 \begin{frame}<1-9> |
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412 \frametitle{\begin{tabular}{c}Alpha-Equivalence\end{tabular}} |
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413 \mbox{}\\[-3mm] |
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414 |
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415 \begin{itemize} |
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416 \item lets first look at pairs\bigskip\medskip |
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417 |
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418 \begin{tabular}{@ {\hspace{1cm}}l} |
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419 $(as, x) \onslide<2->{\approx\!}\makebox[0mm][l]{\only<2-7>{${}_{\text{set}}$}% |
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420 \only<8>{${}_{\text{\alert{list}}}$}% |
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421 \only<9>{${}_{\text{\alert{res}}}$}}% |
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422 \onslide<3->{^{R,\text{fv}}}\,\onslide<2->{(bs,y)}$ |
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423 \end{tabular}\bigskip |
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424 \end{itemize} |
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425 |
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426 \only<1>{ |
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427 \begin{textblock}{8}(3,8.5) |
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428 \begin{tabular}{l@ {\hspace{2mm}}p{8cm}} |
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429 & $as$ is a set of atoms\ldots the binders\\ |
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430 & $x$ is the body\\ |
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431 & $\approx_{\text{set}}$ is where the cardinality |
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432 of the binders has to be the same\\ |
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433 \end{tabular} |
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434 \end{textblock}} |
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435 |
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436 \only<4->{ |
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437 \begin{textblock}{12}(5,8) |
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438 \begin{tabular}{ll@ {\hspace{1mm}}l} |
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439 $\dn$ & \onslide<5->{$\exists \pi.\,$} & $\text{fv}(x) - as = \text{fv}(y) - bs$\\[1mm] |
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440 & \onslide<5->{$\;\;\;\wedge$} & \onslide<5->{$\text{fv}(x) - as \fresh^* \pi$}\\[1mm] |
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441 & \onslide<6->{$\;\;\;\wedge$} & \onslide<6->{$(\pi \act x)\;R\;y$}\\[1mm] |
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442 & \onslide<7-8>{$\;\;\;\wedge$} & \onslide<7-8>{$\pi \act as = bs$}\\ |
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443 \end{tabular} |
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444 \end{textblock}} |
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445 |
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446 \only<8>{ |
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447 \begin{textblock}{8}(3,13.8) |
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448 \footnotesize $^*$ $as$ and $bs$ are \alert{lists} of atoms |
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449 \end{textblock}} |
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450 \end{frame}} |
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451 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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452 *} |
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453 |
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454 text_raw {* |
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455 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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456 \mode<presentation>{ |
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457 \begin{frame}<1-2> |
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458 \frametitle{\begin{tabular}{c}Examples\end{tabular}} |
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459 \mbox{}\\[-3mm] |
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460 |
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461 \begin{itemize} |
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462 \item lets look at ``type-schemes'':\medskip\medskip |
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463 |
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464 \begin{center} |
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465 $(as, x) \approx\!\makebox[0mm][l]{${}_{\text{set}}$}\only<1>{{}^{R,\text{fv}}}\only<2->{{}^{\alert{=},\alert{\text{fv}}}} (bs, y)$ |
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466 \end{center}\medskip |
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467 |
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468 \onslide<2->{ |
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469 \begin{center} |
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470 \begin{tabular}{l} |
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471 $\text{fv}(x) = \{x\}$\\[1mm] |
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472 $\text{fv}(T_1 \rightarrow T_2) = \text{fv}(T_1) \cup \text{fv}(T_2)$\\ |
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473 \end{tabular} |
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474 \end{center}} |
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475 \end{itemize} |
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476 |
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477 |
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478 \only<2->{ |
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479 \begin{textblock}{4}(0.3,12) |
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480 \begin{tikzpicture} |
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481 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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482 {\tiny\color{darkgray} |
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483 \begin{minipage}{3.4cm}\raggedright |
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484 \begin{tabular}{r@ {\hspace{1mm}}l} |
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485 \multicolumn{2}{@ {}l}{res:}\\ |
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486 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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487 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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488 $\wedge$ & $\pi \cdot x = y$\\ |
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489 \\ |
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490 \end{tabular} |
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491 \end{minipage}}; |
601 \end{minipage}}; |
492 \end{tikzpicture} |
602 \end{tikzpicture} |
493 \end{textblock}} |
603 \end{textblock}} |
494 \only<2->{ |
604 |
495 \begin{textblock}{4}(5.2,12) |
605 \end{frame}} |
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606 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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607 *} |
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608 |
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609 text_raw {* |
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610 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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611 \mode<presentation>{ |
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612 \begin{frame}<1-9> |
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613 \frametitle{Substitution} |
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614 |
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615 \begin{tabular}{l@ {\hspace{8mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l@ {}} |
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616 \pgfuseshading{smallbluesphere} & |
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617 \smath{\sigma(\abst{a}{t})} & \smath{\dn} & \smath{\abst{a}{\sigma(t)}}\\[2mm] |
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618 |
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619 \pgfuseshading{smallbluesphere} & |
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620 \smath{\sigma(\pi\act X)} & \smath{\dn} & |
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621 \smath{\begin{cases}% |
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622 \pi\;\act\;\sigma(X) & \!\!\text{if\ } \sigma(X)\not=X\\ |
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623 \pi\act X & \!\!\text{otherwise}% |
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624 \end{cases}}\\[6mm] |
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625 \end{tabular}\bigskip\bigskip |
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626 |
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627 \pause |
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628 \only<2-5>{ |
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629 \only<2->{for example} |
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630 \def\arraystretch{1.3} |
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631 \begin{tabular}{@ {\hspace{14mm}}l@ {\hspace{3mm}}l} |
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632 \onslide<2->{\textcolor{white}{$\Rightarrow$}} & |
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633 \onslide<2->{\alt<3>{\smath{\underline{\abst{a}{\swap{a}{b}\act X}\;\,[X:=\pair{b}{Y}]}}} |
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634 {\smath{\abst{a}{\swap{a}{b}\act X}\;\,[X:=\pair{b}{Y}]}}}\\ |
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635 \onslide<3->{\smath{\Rightarrow}} & |
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636 \onslide<3->{\alt<3,4>{\smath{\abst{a}{\underline{\swap{a}{b}\act X[X:=\pair{b}{Y}]}}}} |
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637 {\smath{\abst{a}{\swap{a}{b}\act X}[X:=\pair{b}{Y}]}}}\\ |
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638 \onslide<4->{\smath{\Rightarrow}} & |
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639 \onslide<4->{\alt<4>{\smath{\abst{a}{\swap{a}{b}\act \underline{\pair{b}{Y}}}}} |
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640 {\smath{\abst{a}{\underline{\swap{a}{b}}\act \pair{b}{Y}}}}}\\ |
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641 \onslide<5->{\smath{\Rightarrow}} & |
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642 \onslide<5->{\smath{\abst{a}{\pair{a}{\swap{a}{b}\act Y}}}} |
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643 \end{tabular}} |
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644 |
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645 \only<6-> |
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646 {\begin{tabular}{l@ {\hspace{8mm}}l@ {}} |
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647 \pgfuseshading{smallbluesphere} & |
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648 if \smath{\nabla\vdash t\approx t'} and\hspace{-2mm}\mbox{} |
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649 \raisebox{-2.7mm}{ |
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650 \alt<7>{\begin{tikzpicture} |
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651 \draw (0,0) node[inner sep=1mm,fill=cream, very thick, draw=red, rounded corners=3mm] |
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652 {\smath{\;\nabla'\vdash\sigma(\nabla)\;}}; |
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653 \end{tikzpicture}} |
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654 {\begin{tikzpicture} |
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655 \draw (0,0) node[inner sep=1mm,fill=white, very thick, draw=white, rounded corners=3mm] |
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656 {\smath{\;\nabla'\vdash\sigma(\nabla)\;}}; |
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657 \end{tikzpicture}}}\\ |
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658 & then \smath{\nabla'\vdash\sigma(t)\approx\sigma(t')} |
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659 \end{tabular}} |
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660 |
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661 \only<9> |
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662 {\begin{tabular}{l@ {\hspace{8mm}}l@ {}} |
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663 \\[-4mm] |
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664 \pgfuseshading{smallbluesphere} & |
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665 \smath{\sigma(\pi\act t)=\pi\act\sigma(t)} |
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666 \end{tabular}} |
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667 |
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668 |
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669 \only<7>{ |
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670 \begin{textblock}{6}(10,10.5) |
496 \begin{tikzpicture} |
671 \begin{tikzpicture} |
497 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
672 \draw (0,0) node[inner sep=1mm,fill=cream, very thick, draw=red, rounded corners=2mm] |
498 {\tiny\color{darkgray} |
673 {\color{darkgray} |
499 \begin{minipage}{3.4cm}\raggedright |
674 \begin{minipage}{3.8cm}\raggedright |
500 \begin{tabular}{r@ {\hspace{1mm}}l} |
675 this means\\[1mm] |
501 \multicolumn{2}{@ {}l}{set:}\\ |
676 \smath{\nabla'\vdash a\fresh\sigma(X)}\\[1mm] |
502 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
677 holds for all\\[1mm] |
503 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
678 \smath{(a\fresh X)\in\nabla} |
504 $\wedge$ & $\pi \cdot x = y$\\ |
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505 $\wedge$ & $\pi \cdot as = bs$\\ |
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506 \end{tabular} |
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507 \end{minipage}}; |
679 \end{minipage}}; |
508 \end{tikzpicture} |
680 \end{tikzpicture} |
509 \end{textblock}} |
681 \end{textblock}} |
510 \only<2->{ |
682 |
511 \begin{textblock}{4}(10.2,12) |
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512 \begin{tikzpicture} |
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513 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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514 {\tiny\color{darkgray} |
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515 \begin{minipage}{3.4cm}\raggedright |
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516 \begin{tabular}{r@ {\hspace{1mm}}l} |
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517 \multicolumn{2}{@ {}l}{list:}\\ |
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518 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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519 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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520 $\wedge$ & $\pi \cdot x = y$\\ |
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521 $\wedge$ & $\pi \cdot as = bs$\\ |
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522 \end{tabular} |
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523 \end{minipage}}; |
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524 \end{tikzpicture} |
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525 \end{textblock}} |
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526 |
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527 \end{frame}} |
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528 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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529 *} |
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530 |
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531 text_raw {* |
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532 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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533 \mode<presentation>{ |
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534 \begin{frame}<1-2> |
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535 \frametitle{\begin{tabular}{c}Examples\end{tabular}} |
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536 \mbox{}\\[-3mm] |
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537 |
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538 \begin{center} |
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539 \only<1>{$(\{x, y\}, x \rightarrow y) \approx_? (\{x, y\}, y \rightarrow x)$} |
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540 \only<2>{$([x, y], x \rightarrow y) \approx_? ([x, y], y \rightarrow x)$} |
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541 \end{center} |
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542 |
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543 \begin{itemize} |
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544 \item $\approx_{\text{res}}$, $\approx_{\text{set}}$% |
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545 \only<2>{, \alert{$\not\approx_{\text{list}}$}} |
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546 \end{itemize} |
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547 |
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548 |
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549 \only<1->{ |
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550 \begin{textblock}{4}(0.3,12) |
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551 \begin{tikzpicture} |
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552 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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553 {\tiny\color{darkgray} |
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555 \begin{tabular}{r@ {\hspace{1mm}}l} |
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556 \multicolumn{2}{@ {}l}{res:}\\ |
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557 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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558 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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559 $\wedge$ & $\pi \cdot x = y$\\ |
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560 \\ |
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561 \end{tabular} |
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562 \end{minipage}}; |
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563 \end{tikzpicture} |
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564 \end{textblock}} |
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565 \only<1->{ |
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568 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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569 {\tiny\color{darkgray} |
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570 \begin{minipage}{3.4cm}\raggedright |
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571 \begin{tabular}{r@ {\hspace{1mm}}l} |
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572 \multicolumn{2}{@ {}l}{set:}\\ |
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573 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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574 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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575 $\wedge$ & $\pi \cdot x = y$\\ |
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576 $\wedge$ & $\pi \cdot as = bs$\\ |
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577 \end{tabular} |
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578 \end{minipage}}; |
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579 \end{tikzpicture} |
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580 \end{textblock}} |
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581 \only<1->{ |
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582 \begin{textblock}{4}(10.2,12) |
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584 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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585 {\tiny\color{darkgray} |
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586 \begin{minipage}{3.4cm}\raggedright |
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587 \begin{tabular}{r@ {\hspace{1mm}}l} |
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588 \multicolumn{2}{@ {}l}{list:}\\ |
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589 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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590 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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591 $\wedge$ & $\pi \cdot x = y$\\ |
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592 $\wedge$ & $\pi \cdot as = bs$\\ |
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593 \end{tabular} |
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594 \end{minipage}}; |
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595 \end{tikzpicture} |
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596 \end{textblock}} |
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597 |
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598 \end{frame}} |
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599 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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600 *} |
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601 |
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602 text_raw {* |
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604 \mode<presentation>{ |
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605 \begin{frame}<1> |
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606 \frametitle{\begin{tabular}{c}Examples\end{tabular}} |
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607 \mbox{}\\[-3mm] |
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608 |
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609 \begin{center} |
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610 \only<1>{$(\{x\}, x) \approx_? (\{x, y\}, x)$} |
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611 \end{center} |
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612 |
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613 \begin{itemize} |
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614 \item $\approx_{\text{res}}$, $\not\approx_{\text{set}}$, |
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615 $\not\approx_{\text{list}}$ |
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616 \end{itemize} |
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617 |
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618 |
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619 \only<1->{ |
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620 \begin{textblock}{4}(0.3,12) |
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622 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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623 {\tiny\color{darkgray} |
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624 \begin{minipage}{3.4cm}\raggedright |
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625 \begin{tabular}{r@ {\hspace{1mm}}l} |
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626 \multicolumn{2}{@ {}l}{res:}\\ |
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627 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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628 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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629 $\wedge$ & $\pi \cdot x = y$\\ |
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630 \\ |
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631 \end{tabular} |
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632 \end{minipage}}; |
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633 \end{tikzpicture} |
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634 \end{textblock}} |
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635 \only<1->{ |
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636 \begin{textblock}{4}(5.2,12) |
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637 \begin{tikzpicture} |
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638 \draw (0,0) node[inner sep=1mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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639 {\tiny\color{darkgray} |
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640 \begin{minipage}{3.4cm}\raggedright |
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641 \begin{tabular}{r@ {\hspace{1mm}}l} |
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642 \multicolumn{2}{@ {}l}{set:}\\ |
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643 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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644 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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645 $\wedge$ & $\pi \cdot x = y$\\ |
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646 $\wedge$ & $\pi \cdot as = bs$\\ |
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647 \end{tabular} |
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648 \end{minipage}}; |
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649 \end{tikzpicture} |
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650 \end{textblock}} |
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651 \only<1->{ |
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652 \begin{textblock}{4}(10.2,12) |
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655 {\tiny\color{darkgray} |
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656 \begin{minipage}{3.4cm}\raggedright |
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657 \begin{tabular}{r@ {\hspace{1mm}}l} |
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658 \multicolumn{2}{@ {}l}{list:}\\ |
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659 $\exists\pi.$ & $\text{fv}(x) - as = \text{fv}(y) - bs$\\ |
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660 $\wedge$ & $\text{fv}(x) - as \fresh^* \pi$\\ |
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661 $\wedge$ & $\pi \cdot x = y$\\ |
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662 $\wedge$ & $\pi \cdot as = bs$\\ |
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663 \end{tabular} |
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664 \end{minipage}}; |
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665 \end{tikzpicture} |
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666 \end{textblock}} |
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667 |
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668 \end{frame}} |
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670 *} |
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671 |
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672 text_raw {* |
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673 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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674 \mode<presentation>{ |
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675 \begin{frame}<1-3> |
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676 \frametitle{\begin{tabular}{c}General Abstractions\end{tabular}} |
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677 \mbox{}\\[-7mm] |
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678 |
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679 \begin{itemize} |
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680 \item we take $(as, x) \approx\!\makebox[0mm][l]{${}_{\star}$}^{=,\text{supp}} (bs, y)$\medskip |
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681 \item they are equivalence relations\medskip |
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682 \item we can therefore use the quotient package to introduce the |
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683 types $\beta\;\text{abs}_\star$\bigskip |
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684 \begin{center} |
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685 \only<1>{$[as].\,x$} |
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686 \only<2>{$\text{supp}([as].x) = \text{supp}(x) - as$} |
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687 \only<3>{% |
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688 \begin{tabular}{r@ {\hspace{1mm}}l} |
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689 \multicolumn{2}{@ {\hspace{-7mm}}l}{$[as]. x \alert{=} [bs].y\;\;\;\text{if\!f}$}\\[2mm] |
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690 $\exists \pi.$ & $\text{supp}(x) - as = \text{supp}(y) - bs$\\ |
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691 $\wedge$ & $\text{supp}(x) - as \fresh^* \pi$\\ |
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692 $\wedge$ & $\pi \act x = y $\\ |
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693 $(\wedge$ & $\pi \act as = bs)\;^\star$\\ |
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694 \end{tabular}} |
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695 \end{center} |
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696 \end{itemize} |
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697 |
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698 |
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699 \end{frame}} |
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700 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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701 *} |
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702 |
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703 text_raw {* |
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704 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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705 \mode<presentation>{ |
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706 \begin{frame}<1> |
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707 \frametitle{\begin{tabular}{c}One Problem\end{tabular}} |
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708 \mbox{}\\[-3mm] |
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709 |
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710 \begin{center} |
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711 $\text{let}\;x_1=t_1 \ldots x_n=t_n\;\text{in}\;s$ |
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712 \end{center} |
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713 |
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714 \begin{itemize} |
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715 \item we cannot represent this as\medskip |
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716 \begin{center} |
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717 $\text{let}\;[x_1,\ldots,x_n]\alert{.}s\;\;[t_1,\ldots,t_n]$ |
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718 \end{center}\bigskip |
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719 |
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720 because\medskip |
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721 \begin{center} |
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722 $\text{let}\;[x].s\;\;[t_1,t_2]$ |
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723 \end{center} |
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724 \end{itemize} |
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725 |
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726 |
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727 \end{frame}} |
683 \end{frame}} |
728 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
684 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
729 *} |
685 *} |
730 |
686 |
731 text_raw {* |
687 text_raw {* |
732 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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733 \mode<presentation>{ |
689 \mode<presentation>{ |
734 \begin{frame}<1-> |
690 \begin{frame}<1-> |
735 \frametitle{\begin{tabular}{c}Our Specifications\end{tabular}} |
691 \frametitle{Equational Problems} |
736 \mbox{}\\[-6mm] |
692 |
737 |
693 An equational problem |
738 \mbox{}\hspace{10mm} |
694 \[ |
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695 \colorbox{cream}{\smath{t \eqprob t'}} |
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696 \] |
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697 is \alert{solved} by |
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698 |
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699 \begin{center} |
739 \begin{tabular}{ll} |
700 \begin{tabular}{ll} |
740 \multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\ |
701 \pgfuseshading{smallbluesphere} & a substitution \smath{\sigma} (terms for variables)\\[3mm] |
741 \hspace{5mm}\phantom{$|$} Var name\\ |
702 \pgfuseshading{smallbluesphere} & {\bf and} a set of freshness assumptions \smath{\nabla} |
742 \hspace{5mm}$|$ App trm trm\\ |
703 \end{tabular} |
743 \hspace{5mm}$|$ Lam x::name t::trm |
704 \end{center} |
744 & \isacommand{bind} x \isacommand{in} t\\ |
705 |
745 \hspace{5mm}$|$ Let as::assn t::trm |
706 so that \smath{\nabla\vdash \sigma(t)\approx \sigma(t')}. |
746 & \isacommand{bind} bn(as) \isacommand{in} t\\ |
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747 \multicolumn{2}{l}{\isacommand{and} assn $=$}\\ |
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748 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\ |
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749 \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\ |
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750 \multicolumn{2}{l}{\isacommand{binder} bn \isacommand{where}}\\ |
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751 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\ |
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752 \multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\ |
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753 \end{tabular} |
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754 |
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755 \end{frame}} |
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756 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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757 *} |
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758 |
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759 text_raw {* |
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760 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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761 \mode<presentation>{ |
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762 \begin{frame}<1-2> |
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763 \frametitle{\begin{tabular}{c}``Raw'' Definitions\end{tabular}} |
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764 \mbox{}\\[-6mm] |
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765 |
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766 \mbox{}\hspace{10mm} |
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767 \begin{tabular}{ll} |
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768 \multicolumn{2}{l}{\isacommand{datatype} trm $=$}\\ |
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769 \hspace{5mm}\phantom{$|$} Var name\\ |
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770 \hspace{5mm}$|$ App trm trm\\ |
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771 \hspace{5mm}$|$ Lam name trm\\ |
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772 \hspace{5mm}$|$ Let assn trm\\ |
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773 \multicolumn{2}{l}{\isacommand{and} assn $=$}\\ |
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774 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\ |
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775 \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\[5mm] |
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776 \multicolumn{2}{l}{\isacommand{function} bn \isacommand{where}}\\ |
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777 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\ |
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778 \multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\ |
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779 \end{tabular} |
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780 |
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781 \only<2>{ |
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782 \begin{textblock}{5}(10,5) |
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783 $+$ \begin{tabular}{l}automatically\\ |
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784 generate fv's\end{tabular} |
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785 \end{textblock}} |
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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}} |
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795 \mbox{}\\[6mm] |
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796 |
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797 \begin{center} |
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798 Lam x::name t::trm \hspace{10mm}\isacommand{bind} x \isacommand{in} t\\ |
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799 \end{center} |
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800 |
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801 |
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802 \[ |
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803 \infer[\text{Lam-}\!\approx_\alpha] |
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804 {\text{Lam}\;x\;t \approx_\alpha \text{Lam}\;x'\;t'} |
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805 {([x], t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$} |
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806 ^{\approx_\alpha,\text{fv}} ([x'], t')} |
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807 \] |
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808 |
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809 |
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810 \end{frame}} |
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811 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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812 *} |
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813 |
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814 text_raw {* |
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815 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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816 \mode<presentation>{ |
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817 \begin{frame}<1> |
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818 \frametitle{\begin{tabular}{c}\LARGE``Raw'' Alpha-Equivalence\end{tabular}} |
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819 \mbox{}\\[6mm] |
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820 |
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821 \begin{center} |
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822 Lam x::name y::name t::trm s::trm \hspace{5mm}\isacommand{bind} x y \isacommand{in} t s\\ |
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823 \end{center} |
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824 |
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825 |
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826 \[ |
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827 \infer[\text{Lam-}\!\approx_\alpha] |
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828 {\text{Lam}\;x\;y\;t\;s \approx_\alpha \text{Lam}\;x'\;y'\;t'\;s'} |
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829 {([x, y], (t, s)) \approx\!\makebox[0mm][l]{${}_{\text{list}}$} |
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830 ^{R, fv} ([x', y'], (t', s'))} |
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831 \] |
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832 |
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833 \footnotesize |
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834 where $R =\;\approx_\alpha\times\approx_\alpha$ and $fv = \text{fv}\cup\text{fv}$ |
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835 |
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836 \end{frame}} |
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837 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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838 *} |
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839 |
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840 text_raw {* |
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841 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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842 \mode<presentation>{ |
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843 \begin{frame}<1-2> |
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844 \frametitle{\begin{tabular}{c}\LARGE``Raw'' Alpha-Equivalence\end{tabular}} |
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845 \mbox{}\\[6mm] |
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846 |
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847 \begin{center} |
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848 Let as::assn t::trm \hspace{10mm}\isacommand{bind} bn(as) \isacommand{in} t\\ |
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849 \end{center} |
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850 |
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851 |
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852 \[ |
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853 \infer[\text{Let-}\!\approx_\alpha] |
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854 {\text{Let}\;as\;t \approx_\alpha \text{Let}\;as'\;t'} |
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855 {(\text{bn}(as), t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$} |
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856 ^{\approx_\alpha,\text{fv}} (\text{bn}(as'), t') & |
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857 \onslide<2>{as \approx_\alpha^{\text{bn}} as'}} |
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858 \] |
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859 |
707 |
860 |
708 |
861 \end{frame}} |
709 \end{frame}} |
862 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
710 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
863 *} |
711 *} |
864 |
712 |
865 text_raw {* |
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867 \mode<presentation>{ |
715 \mode<presentation>{ |
868 \begin{frame}<1-> |
716 \begin{frame}<1-> |
869 \frametitle{\begin{tabular}{c}\LARGE{}$\alpha$ for Binding Functions\end{tabular}} |
717 \frametitle{Conclusion} |
870 \mbox{}\\[-6mm] |
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871 |
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872 \mbox{}\hspace{10mm} |
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873 \begin{tabular}{l} |
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874 \ldots\\ |
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875 \isacommand{binder} bn \isacommand{where}\\ |
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876 \phantom{$|$} bn(ANil) $=$ $[]$\\ |
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877 $|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)\\ |
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878 \end{tabular}\bigskip |
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879 |
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880 \begin{center} |
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881 \mbox{\infer{\text{ANil} \approx_\alpha^{\text{bn}} \text{ANil}}{}}\bigskip |
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882 |
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883 \mbox{\infer{\text{ACons}\;a\;t\;as \approx_\alpha^{\text{bn}} \text{ACons}\;a'\;t'\;as'} |
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884 {t \approx_\alpha t' & as \approx_\alpha^{bn} as'}} |
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885 \end{center} |
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886 |
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887 |
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888 \end{frame}} |
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889 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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890 *} |
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893 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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894 \mode<presentation>{ |
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895 \begin{frame}<1-> |
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896 \frametitle{\begin{tabular}{c}Automatic Proofs\end{tabular}} |
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897 \mbox{}\\[-6mm] |
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898 |
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899 \begin{itemize} |
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900 \item we can show that $\alpha$'s are equivalence relations\medskip |
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901 \item as a result we can use the quotient package to introduce the type(s) |
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902 of $\alpha$-equated terms |
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903 |
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904 \[ |
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905 \infer |
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906 {\text{Lam}\;x\;t \alert{=} \text{Lam}\;x'\;t'} |
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907 {\only<1>{([x], t) \approx\!\makebox[0mm][l]{${}_{\text{list}}$} |
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908 ^{=,\text{supp}} ([x'], t')}% |
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909 \only<2>{[x].t = [x'].t'}} |
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910 \] |
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911 |
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912 |
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913 \item the properties for support are implied by the properties of $[\_].\_$ |
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914 \item we can derive strong induction principles (almost automatic---just a matter of |
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915 another week or two) |
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916 \end{itemize} |
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917 |
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918 |
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919 \end{frame}} |
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920 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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921 *} |
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922 |
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923 text_raw {* |
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924 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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925 \mode<presentation>{ |
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926 \begin{frame}<1>[t] |
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927 \frametitle{\begin{tabular}{c}Runtime is Acceptable\end{tabular}} |
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928 \mbox{}\\[-7mm]\mbox{} |
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929 |
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930 \footnotesize |
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931 \begin{center} |
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932 \begin{tikzpicture} |
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933 \draw (0,0) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm] |
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934 (A) {\begin{minipage}{0.8cm}bind.\\spec.\end{minipage}}; |
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935 |
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936 \draw (2,0) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm] |
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937 (B) {\begin{minipage}{0.8cm}raw\\terms\end{minipage}}; |
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938 |
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939 \draw (4,0) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm] |
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940 (C) {\begin{minipage}{0.8cm}$\alpha$-\\equiv.\end{minipage}}; |
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941 |
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942 \draw (0,-2) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm] |
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943 (D) {\begin{minipage}{0.8cm}quot.\\type\end{minipage}}; |
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944 |
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945 \draw (2,-2) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm] |
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946 (E) {\begin{minipage}{0.8cm}lift\\thms\end{minipage}}; |
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947 |
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948 \draw (4,-2) node[inner sep=2mm, ultra thick, draw=white, rounded corners=2mm] |
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949 (F) {\begin{minipage}{0.8cm}add.\\thms\end{minipage}}; |
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950 |
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951 \draw[->,white!50,line width=1mm] (A) -- (B); |
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952 \draw[->,white!50,line width=1mm] (B) -- (C); |
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953 \draw[->,white!50,line width=1mm, line join=round, rounded corners=2mm] |
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954 (C) -- (5,0) -- (5,-1) -- (-1,-1) -- (-1,-2) -- (D); |
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955 \draw[->,white!50,line width=1mm] (D) -- (E); |
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956 \draw[->,white!50,line width=1mm] (E) -- (F); |
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957 \end{tikzpicture} |
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958 \end{center} |
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959 |
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960 \begin{itemize} |
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961 \item Core Haskell: 11 types, 49 term-constructors, |
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962 \end{itemize} |
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963 |
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964 \end{frame}} |
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965 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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966 *} |
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967 |
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968 |
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969 text_raw {* |
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970 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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971 \mode<presentation>{ |
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972 \begin{frame}<1-> |
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973 \frametitle{\begin{tabular}{c}Interesting Phenomenon\end{tabular}} |
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974 \mbox{}\\[-6mm] |
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975 |
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976 \small |
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977 \mbox{}\hspace{10mm} |
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978 \begin{tabular}{ll} |
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979 \multicolumn{2}{l}{\isacommand{nominal\_datatype} trm $=$}\\ |
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980 \hspace{5mm}\phantom{$|$} Var name\\ |
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981 \hspace{5mm}$|$ App trm trm\\ |
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982 \hspace{5mm}$|$ Lam x::name t::trm |
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983 & \isacommand{bind} x \isacommand{in} t\\ |
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984 \hspace{5mm}$|$ Let as::assn t::trm |
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985 & \isacommand{bind} bn(as) \isacommand{in} t\\ |
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986 \multicolumn{2}{l}{\isacommand{and} assn $=$}\\ |
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987 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} ANil}\\ |
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988 \multicolumn{2}{l}{\hspace{5mm}$|$ ACons name trm assn}\\ |
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989 \multicolumn{2}{l}{\isacommand{binder} bn \isacommand{where}}\\ |
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990 \multicolumn{2}{l}{\hspace{5mm}\phantom{$|$} bn(ANil) $=$ $[]$}\\ |
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991 \multicolumn{2}{l}{\hspace{5mm}$|$ bn(ACons a t as) $=$ $[$a$]$ @ bn(as)}\\ |
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992 \end{tabular}\bigskip\medskip |
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993 |
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994 we cannot quotient assn: ACons a \ldots $\not\approx_\alpha$ ACons b \ldots |
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995 |
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996 \end{frame}} |
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997 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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998 *} |
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999 |
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1000 text_raw {* |
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1001 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1002 \mode<presentation>{ |
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1003 \begin{frame}<1-> |
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1004 \frametitle{\begin{tabular}{c}Conclusion\end{tabular}} |
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1005 \mbox{}\\[-6mm] |
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1006 |
718 |
1007 \begin{itemize} |
719 \begin{itemize} |
1008 \item the user does not see anything of the raw level\medskip |
720 \item the user does not see anything of the raw level\medskip |
1009 \only<1>{\begin{center} |
721 \only<1>{\begin{center} |
1010 Lam a (Var a) \alert{$=$} Lam b (Var b) |
722 Lam a (Var a) \alert{$=$} Lam b (Var b) |