1 % |
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2 \begin{isabellebody}% |
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3 \def\isabellecontext{prelim}% |
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4 % |
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5 \isadelimtheory |
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6 \isanewline |
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7 \isanewline |
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8 \isanewline |
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9 % |
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10 \endisadelimtheory |
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11 % |
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12 \isatagtheory |
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13 \isacommand{theory}\isamarkupfalse% |
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14 \ prelim\ \isakeyword{imports}\ base\ \isakeyword{begin}% |
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15 \endisatagtheory |
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16 {\isafoldtheory}% |
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17 % |
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18 \isadelimtheory |
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19 % |
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20 \endisadelimtheory |
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21 % |
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22 \isamarkupchapter{Preliminaries% |
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23 } |
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24 \isamarkuptrue% |
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25 % |
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26 \isamarkupsection{Contexts \label{sec:context}% |
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27 } |
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28 \isamarkuptrue% |
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29 % |
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30 \begin{isamarkuptext}% |
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31 A logical context represents the background that is required for |
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32 formulating statements and composing proofs. It acts as a medium to |
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33 produce formal content, depending on earlier material (declarations, |
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34 results etc.). |
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35 |
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36 For example, derivations within the Isabelle/Pure logic can be |
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37 described as a judgment \isa{{\isasymGamma}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\ {\isasymphi}}, which means that a |
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38 proposition \isa{{\isasymphi}} is derivable from hypotheses \isa{{\isasymGamma}} |
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39 within the theory \isa{{\isasymTheta}}. There are logical reasons for |
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40 keeping \isa{{\isasymTheta}} and \isa{{\isasymGamma}} separate: theories can be |
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41 liberal about supporting type constructors and schematic |
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42 polymorphism of constants and axioms, while the inner calculus of |
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43 \isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymphi}} is strictly limited to Simple Type Theory (with |
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44 fixed type variables in the assumptions). |
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45 |
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46 \medskip Contexts and derivations are linked by the following key |
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47 principles: |
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48 |
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49 \begin{itemize} |
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50 |
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51 \item Transfer: monotonicity of derivations admits results to be |
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52 transferred into a \emph{larger} context, i.e.\ \isa{{\isasymGamma}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\ {\isasymphi}} implies \isa{{\isasymGamma}{\isacharprime}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\isactrlsub {\isacharprime}\ {\isasymphi}} for contexts \isa{{\isasymTheta}{\isacharprime}\ {\isasymsupseteq}\ {\isasymTheta}} and \isa{{\isasymGamma}{\isacharprime}\ {\isasymsupseteq}\ {\isasymGamma}}. |
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53 |
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54 \item Export: discharge of hypotheses admits results to be exported |
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55 into a \emph{smaller} context, i.e.\ \isa{{\isasymGamma}{\isacharprime}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\ {\isasymphi}} |
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56 implies \isa{{\isasymGamma}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\ {\isasymDelta}\ {\isasymLongrightarrow}\ {\isasymphi}} where \isa{{\isasymGamma}{\isacharprime}\ {\isasymsupseteq}\ {\isasymGamma}} and |
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57 \isa{{\isasymDelta}\ {\isacharequal}\ {\isasymGamma}{\isacharprime}\ {\isacharminus}\ {\isasymGamma}}. Note that \isa{{\isasymTheta}} remains unchanged here, |
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58 only the \isa{{\isasymGamma}} part is affected. |
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59 |
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60 \end{itemize} |
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61 |
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62 \medskip By modeling the main characteristics of the primitive |
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63 \isa{{\isasymTheta}} and \isa{{\isasymGamma}} above, and abstracting over any |
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64 particular logical content, we arrive at the fundamental notions of |
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65 \emph{theory context} and \emph{proof context} in Isabelle/Isar. |
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66 These implement a certain policy to manage arbitrary \emph{context |
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67 data}. There is a strongly-typed mechanism to declare new kinds of |
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68 data at compile time. |
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69 |
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70 The internal bootstrap process of Isabelle/Pure eventually reaches a |
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71 stage where certain data slots provide the logical content of \isa{{\isasymTheta}} and \isa{{\isasymGamma}} sketched above, but this does not stop there! |
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72 Various additional data slots support all kinds of mechanisms that |
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73 are not necessarily part of the core logic. |
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74 |
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75 For example, there would be data for canonical introduction and |
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76 elimination rules for arbitrary operators (depending on the |
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77 object-logic and application), which enables users to perform |
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78 standard proof steps implicitly (cf.\ the \isa{rule} method |
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79 \cite{isabelle-isar-ref}). |
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80 |
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81 \medskip Thus Isabelle/Isar is able to bring forth more and more |
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82 concepts successively. In particular, an object-logic like |
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83 Isabelle/HOL continues the Isabelle/Pure setup by adding specific |
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84 components for automated reasoning (classical reasoner, tableau |
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85 prover, structured induction etc.) and derived specification |
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86 mechanisms (inductive predicates, recursive functions etc.). All of |
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87 this is ultimately based on the generic data management by theory |
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88 and proof contexts introduced here.% |
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89 \end{isamarkuptext}% |
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90 \isamarkuptrue% |
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91 % |
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92 \isamarkupsubsection{Theory context \label{sec:context-theory}% |
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93 } |
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94 \isamarkuptrue% |
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95 % |
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96 \begin{isamarkuptext}% |
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97 \glossary{Theory}{FIXME} |
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98 |
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99 A \emph{theory} is a data container with explicit named and unique |
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100 identifier. Theories are related by a (nominal) sub-theory |
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101 relation, which corresponds to the dependency graph of the original |
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102 construction; each theory is derived from a certain sub-graph of |
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103 ancestor theories. |
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104 |
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105 The \isa{merge} operation produces the least upper bound of two |
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106 theories, which actually degenerates into absorption of one theory |
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107 into the other (due to the nominal sub-theory relation). |
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108 |
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109 The \isa{begin} operation starts a new theory by importing |
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110 several parent theories and entering a special \isa{draft} mode, |
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111 which is sustained until the final \isa{end} operation. A draft |
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112 theory acts like a linear type, where updates invalidate earlier |
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113 versions. An invalidated draft is called ``stale''. |
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114 |
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115 The \isa{checkpoint} operation produces an intermediate stepping |
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116 stone that will survive the next update: both the original and the |
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117 changed theory remain valid and are related by the sub-theory |
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118 relation. Checkpointing essentially recovers purely functional |
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119 theory values, at the expense of some extra internal bookkeeping. |
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120 |
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121 The \isa{copy} operation produces an auxiliary version that has |
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122 the same data content, but is unrelated to the original: updates of |
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123 the copy do not affect the original, neither does the sub-theory |
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124 relation hold. |
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125 |
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126 \medskip The example in \figref{fig:ex-theory} below shows a theory |
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127 graph derived from \isa{Pure}, with theory \isa{Length} |
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128 importing \isa{Nat} and \isa{List}. The body of \isa{Length} consists of a sequence of updates, working mostly on |
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129 drafts. Intermediate checkpoints may occur as well, due to the |
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130 history mechanism provided by the Isar top-level, cf.\ |
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131 \secref{sec:isar-toplevel}. |
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132 |
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133 \begin{figure}[htb] |
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134 \begin{center} |
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135 \begin{tabular}{rcccl} |
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136 & & \isa{Pure} \\ |
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137 & & \isa{{\isasymdown}} \\ |
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138 & & \isa{FOL} \\ |
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139 & $\swarrow$ & & $\searrow$ & \\ |
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140 \isa{Nat} & & & & \isa{List} \\ |
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141 & $\searrow$ & & $\swarrow$ \\ |
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142 & & \isa{Length} \\ |
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143 & & \multicolumn{3}{l}{~~\hyperlink{keyword.imports}{\mbox{\isa{\isakeyword{imports}}}}} \\ |
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144 & & \multicolumn{3}{l}{~~\hyperlink{keyword.begin}{\mbox{\isa{\isakeyword{begin}}}}} \\ |
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145 & & $\vdots$~~ \\ |
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146 & & \isa{{\isasymbullet}}~~ \\ |
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147 & & $\vdots$~~ \\ |
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148 & & \isa{{\isasymbullet}}~~ \\ |
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149 & & $\vdots$~~ \\ |
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150 & & \multicolumn{3}{l}{~~\hyperlink{command.end}{\mbox{\isa{\isacommand{end}}}}} \\ |
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151 \end{tabular} |
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152 \caption{A theory definition depending on ancestors}\label{fig:ex-theory} |
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153 \end{center} |
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154 \end{figure} |
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155 |
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156 \medskip There is a separate notion of \emph{theory reference} for |
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157 maintaining a live link to an evolving theory context: updates on |
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158 drafts are propagated automatically. Dynamic updating stops after |
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159 an explicit \isa{end} only. |
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160 |
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161 Derived entities may store a theory reference in order to indicate |
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162 the context they belong to. This implicitly assumes monotonic |
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163 reasoning, because the referenced context may become larger without |
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164 further notice.% |
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165 \end{isamarkuptext}% |
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166 \isamarkuptrue% |
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167 % |
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168 \isadelimmlref |
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169 % |
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170 \endisadelimmlref |
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171 % |
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172 \isatagmlref |
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173 % |
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174 \begin{isamarkuptext}% |
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175 \begin{mldecls} |
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176 \indexmltype{theory}\verb|type theory| \\ |
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177 \indexml{Theory.subthy}\verb|Theory.subthy: theory * theory -> bool| \\ |
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178 \indexml{Theory.merge}\verb|Theory.merge: theory * theory -> theory| \\ |
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179 \indexml{Theory.checkpoint}\verb|Theory.checkpoint: theory -> theory| \\ |
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180 \indexml{Theory.copy}\verb|Theory.copy: theory -> theory| \\ |
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181 \end{mldecls} |
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182 \begin{mldecls} |
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183 \indexmltype{theory\_ref}\verb|type theory_ref| \\ |
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184 \indexml{Theory.deref}\verb|Theory.deref: theory_ref -> theory| \\ |
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185 \indexml{Theory.check\_thy}\verb|Theory.check_thy: theory -> theory_ref| \\ |
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186 \end{mldecls} |
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187 |
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188 \begin{description} |
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189 |
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190 \item \verb|theory| represents theory contexts. This is |
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191 essentially a linear type! Most operations destroy the original |
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192 version, which then becomes ``stale''. |
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193 |
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194 \item \verb|Theory.subthy|~\isa{{\isacharparenleft}thy\isactrlsub {\isadigit{1}}{\isacharcomma}\ thy\isactrlsub {\isadigit{2}}{\isacharparenright}} |
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195 compares theories according to the inherent graph structure of the |
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196 construction. This sub-theory relation is a nominal approximation |
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197 of inclusion (\isa{{\isasymsubseteq}}) of the corresponding content. |
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198 |
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199 \item \verb|Theory.merge|~\isa{{\isacharparenleft}thy\isactrlsub {\isadigit{1}}{\isacharcomma}\ thy\isactrlsub {\isadigit{2}}{\isacharparenright}} |
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200 absorbs one theory into the other. This fails for unrelated |
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201 theories! |
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202 |
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203 \item \verb|Theory.checkpoint|~\isa{thy} produces a safe |
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204 stepping stone in the linear development of \isa{thy}. The next |
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205 update will result in two related, valid theories. |
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206 |
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207 \item \verb|Theory.copy|~\isa{thy} produces a variant of \isa{thy} that holds a copy of the same data. The result is not |
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208 related to the original; the original is unchanched. |
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209 |
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210 \item \verb|theory_ref| represents a sliding reference to an |
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211 always valid theory; updates on the original are propagated |
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212 automatically. |
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213 |
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214 \item \verb|Theory.deref|~\isa{thy{\isacharunderscore}ref} turns a \verb|theory_ref| into an \verb|theory| value. As the referenced |
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215 theory evolves monotonically over time, later invocations of \verb|Theory.deref| may refer to a larger context. |
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216 |
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217 \item \verb|Theory.check_thy|~\isa{thy} produces a \verb|theory_ref| from a valid \verb|theory| value. |
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218 |
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219 \end{description}% |
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220 \end{isamarkuptext}% |
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221 \isamarkuptrue% |
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222 % |
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223 \endisatagmlref |
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224 {\isafoldmlref}% |
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225 % |
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226 \isadelimmlref |
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227 % |
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228 \endisadelimmlref |
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229 % |
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230 \isamarkupsubsection{Proof context \label{sec:context-proof}% |
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231 } |
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232 \isamarkuptrue% |
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233 % |
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234 \begin{isamarkuptext}% |
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235 \glossary{Proof context}{The static context of a structured proof, |
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236 acts like a local ``theory'' of the current portion of Isar proof |
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237 text, generalizes the idea of local hypotheses \isa{{\isasymGamma}} in |
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238 judgments \isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymphi}} of natural deduction calculi. There is a |
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239 generic notion of introducing and discharging hypotheses. |
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240 Arbritrary auxiliary context data may be adjoined.} |
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241 |
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242 A proof context is a container for pure data with a back-reference |
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243 to the theory it belongs to. The \isa{init} operation creates a |
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244 proof context from a given theory. Modifications to draft theories |
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245 are propagated to the proof context as usual, but there is also an |
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246 explicit \isa{transfer} operation to force resynchronization |
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247 with more substantial updates to the underlying theory. The actual |
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248 context data does not require any special bookkeeping, thanks to the |
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249 lack of destructive features. |
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250 |
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251 Entities derived in a proof context need to record inherent logical |
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252 requirements explicitly, since there is no separate context |
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253 identification as for theories. For example, hypotheses used in |
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254 primitive derivations (cf.\ \secref{sec:thms}) are recorded |
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255 separately within the sequent \isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymphi}}, just to make double |
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256 sure. Results could still leak into an alien proof context do to |
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257 programming errors, but Isabelle/Isar includes some extra validity |
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258 checks in critical positions, notably at the end of a sub-proof. |
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259 |
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260 Proof contexts may be manipulated arbitrarily, although the common |
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261 discipline is to follow block structure as a mental model: a given |
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262 context is extended consecutively, and results are exported back |
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263 into the original context. Note that the Isar proof states model |
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264 block-structured reasoning explicitly, using a stack of proof |
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265 contexts internally, cf.\ \secref{sec:isar-proof-state}.% |
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266 \end{isamarkuptext}% |
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267 \isamarkuptrue% |
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268 % |
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269 \isadelimmlref |
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270 % |
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271 \endisadelimmlref |
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272 % |
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273 \isatagmlref |
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274 % |
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275 \begin{isamarkuptext}% |
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276 \begin{mldecls} |
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277 \indexmltype{Proof.context}\verb|type Proof.context| \\ |
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278 \indexml{ProofContext.init}\verb|ProofContext.init: theory -> Proof.context| \\ |
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279 \indexml{ProofContext.theory\_of}\verb|ProofContext.theory_of: Proof.context -> theory| \\ |
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280 \indexml{ProofContext.transfer}\verb|ProofContext.transfer: theory -> Proof.context -> Proof.context| \\ |
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281 \end{mldecls} |
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282 |
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283 \begin{description} |
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284 |
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285 \item \verb|Proof.context| represents proof contexts. Elements |
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286 of this type are essentially pure values, with a sliding reference |
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287 to the background theory. |
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288 |
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289 \item \verb|ProofContext.init|~\isa{thy} produces a proof context |
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290 derived from \isa{thy}, initializing all data. |
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291 |
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292 \item \verb|ProofContext.theory_of|~\isa{ctxt} selects the |
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293 background theory from \isa{ctxt}, dereferencing its internal |
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294 \verb|theory_ref|. |
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295 |
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296 \item \verb|ProofContext.transfer|~\isa{thy\ ctxt} promotes the |
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297 background theory of \isa{ctxt} to the super theory \isa{thy}. |
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298 |
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299 \end{description}% |
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300 \end{isamarkuptext}% |
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301 \isamarkuptrue% |
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302 % |
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303 \endisatagmlref |
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304 {\isafoldmlref}% |
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305 % |
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306 \isadelimmlref |
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307 % |
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308 \endisadelimmlref |
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309 % |
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310 \isamarkupsubsection{Generic contexts \label{sec:generic-context}% |
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311 } |
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312 \isamarkuptrue% |
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313 % |
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314 \begin{isamarkuptext}% |
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315 A generic context is the disjoint sum of either a theory or proof |
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316 context. Occasionally, this enables uniform treatment of generic |
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317 context data, typically extra-logical information. Operations on |
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318 generic contexts include the usual injections, partial selections, |
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319 and combinators for lifting operations on either component of the |
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320 disjoint sum. |
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321 |
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322 Moreover, there are total operations \isa{theory{\isacharunderscore}of} and \isa{proof{\isacharunderscore}of} to convert a generic context into either kind: a theory |
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323 can always be selected from the sum, while a proof context might |
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324 have to be constructed by an ad-hoc \isa{init} operation.% |
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325 \end{isamarkuptext}% |
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326 \isamarkuptrue% |
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327 % |
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328 \isadelimmlref |
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329 % |
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330 \endisadelimmlref |
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331 % |
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332 \isatagmlref |
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333 % |
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334 \begin{isamarkuptext}% |
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335 \begin{mldecls} |
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336 \indexmltype{Context.generic}\verb|type Context.generic| \\ |
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337 \indexml{Context.theory\_of}\verb|Context.theory_of: Context.generic -> theory| \\ |
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338 \indexml{Context.proof\_of}\verb|Context.proof_of: Context.generic -> Proof.context| \\ |
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339 \end{mldecls} |
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340 |
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341 \begin{description} |
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342 |
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343 \item \verb|Context.generic| is the direct sum of \verb|theory| and \verb|Proof.context|, with the datatype |
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344 constructors \verb|Context.Theory| and \verb|Context.Proof|. |
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345 |
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346 \item \verb|Context.theory_of|~\isa{context} always produces a |
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347 theory from the generic \isa{context}, using \verb|ProofContext.theory_of| as required. |
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348 |
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349 \item \verb|Context.proof_of|~\isa{context} always produces a |
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350 proof context from the generic \isa{context}, using \verb|ProofContext.init| as required (note that this re-initializes the |
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351 context data with each invocation). |
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352 |
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353 \end{description}% |
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354 \end{isamarkuptext}% |
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355 \isamarkuptrue% |
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356 % |
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357 \endisatagmlref |
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358 {\isafoldmlref}% |
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359 % |
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360 \isadelimmlref |
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361 % |
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362 \endisadelimmlref |
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363 % |
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364 \isamarkupsubsection{Context data \label{sec:context-data}% |
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365 } |
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366 \isamarkuptrue% |
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367 % |
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368 \begin{isamarkuptext}% |
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369 The main purpose of theory and proof contexts is to manage arbitrary |
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370 data. New data types can be declared incrementally at compile time. |
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371 There are separate declaration mechanisms for any of the three kinds |
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372 of contexts: theory, proof, generic. |
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373 |
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374 \paragraph{Theory data} may refer to destructive entities, which are |
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375 maintained in direct correspondence to the linear evolution of |
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376 theory values, including explicit copies.\footnote{Most existing |
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377 instances of destructive theory data are merely historical relics |
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378 (e.g.\ the destructive theorem storage, and destructive hints for |
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379 the Simplifier and Classical rules).} A theory data declaration |
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380 needs to implement the following SML signature: |
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381 |
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382 \medskip |
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383 \begin{tabular}{ll} |
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384 \isa{{\isasymtype}\ T} & representing type \\ |
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385 \isa{{\isasymval}\ empty{\isacharcolon}\ T} & empty default value \\ |
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386 \isa{{\isasymval}\ copy{\isacharcolon}\ T\ {\isasymrightarrow}\ T} & refresh impure data \\ |
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387 \isa{{\isasymval}\ extend{\isacharcolon}\ T\ {\isasymrightarrow}\ T} & re-initialize on import \\ |
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388 \isa{{\isasymval}\ merge{\isacharcolon}\ T\ {\isasymtimes}\ T\ {\isasymrightarrow}\ T} & join on import \\ |
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389 \end{tabular} |
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390 \medskip |
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391 |
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392 \noindent The \isa{empty} value acts as initial default for |
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393 \emph{any} theory that does not declare actual data content; \isa{copy} maintains persistent integrity for impure data, it is just |
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394 the identity for pure values; \isa{extend} is acts like a |
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395 unitary version of \isa{merge}, both operations should also |
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396 include the functionality of \isa{copy} for impure data. |
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397 |
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398 \paragraph{Proof context data} is purely functional. A declaration |
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399 needs to implement the following SML signature: |
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400 |
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401 \medskip |
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402 \begin{tabular}{ll} |
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403 \isa{{\isasymtype}\ T} & representing type \\ |
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404 \isa{{\isasymval}\ init{\isacharcolon}\ theory\ {\isasymrightarrow}\ T} & produce initial value \\ |
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405 \end{tabular} |
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406 \medskip |
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407 |
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408 \noindent The \isa{init} operation is supposed to produce a pure |
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409 value from the given background theory. |
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410 |
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411 \paragraph{Generic data} provides a hybrid interface for both theory |
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412 and proof data. The declaration is essentially the same as for |
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413 (pure) theory data, without \isa{copy}. The \isa{init} |
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414 operation for proof contexts merely selects the current data value |
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415 from the background theory. |
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416 |
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417 \bigskip A data declaration of type \isa{T} results in the |
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418 following interface: |
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419 |
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420 \medskip |
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421 \begin{tabular}{ll} |
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422 \isa{init{\isacharcolon}\ theory\ {\isasymrightarrow}\ theory} \\ |
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423 \isa{get{\isacharcolon}\ context\ {\isasymrightarrow}\ T} \\ |
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424 \isa{put{\isacharcolon}\ T\ {\isasymrightarrow}\ context\ {\isasymrightarrow}\ context} \\ |
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425 \isa{map{\isacharcolon}\ {\isacharparenleft}T\ {\isasymrightarrow}\ T{\isacharparenright}\ {\isasymrightarrow}\ context\ {\isasymrightarrow}\ context} \\ |
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426 \end{tabular} |
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427 \medskip |
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428 |
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429 \noindent Here \isa{init} is only applicable to impure theory |
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430 data to install a fresh copy persistently (destructive update on |
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431 uninitialized has no permanent effect). The other operations provide |
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432 access for the particular kind of context (theory, proof, or generic |
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433 context). Note that this is a safe interface: there is no other way |
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434 to access the corresponding data slot of a context. By keeping |
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435 these operations private, a component may maintain abstract values |
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436 authentically, without other components interfering.% |
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437 \end{isamarkuptext}% |
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438 \isamarkuptrue% |
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439 % |
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440 \isadelimmlref |
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441 % |
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442 \endisadelimmlref |
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443 % |
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444 \isatagmlref |
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445 % |
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446 \begin{isamarkuptext}% |
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447 \begin{mldecls} |
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448 \indexmlfunctor{TheoryDataFun}\verb|functor TheoryDataFun| \\ |
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449 \indexmlfunctor{ProofDataFun}\verb|functor ProofDataFun| \\ |
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450 \indexmlfunctor{GenericDataFun}\verb|functor GenericDataFun| \\ |
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451 \end{mldecls} |
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452 |
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453 \begin{description} |
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454 |
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455 \item \verb|TheoryDataFun|\isa{{\isacharparenleft}spec{\isacharparenright}} declares data for |
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456 type \verb|theory| according to the specification provided as |
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457 argument structure. The resulting structure provides data init and |
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458 access operations as described above. |
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459 |
|
460 \item \verb|ProofDataFun|\isa{{\isacharparenleft}spec{\isacharparenright}} is analogous to |
|
461 \verb|TheoryDataFun| for type \verb|Proof.context|. |
|
462 |
|
463 \item \verb|GenericDataFun|\isa{{\isacharparenleft}spec{\isacharparenright}} is analogous to |
|
464 \verb|TheoryDataFun| for type \verb|Context.generic|. |
|
465 |
|
466 \end{description}% |
|
467 \end{isamarkuptext}% |
|
468 \isamarkuptrue% |
|
469 % |
|
470 \endisatagmlref |
|
471 {\isafoldmlref}% |
|
472 % |
|
473 \isadelimmlref |
|
474 % |
|
475 \endisadelimmlref |
|
476 % |
|
477 \isamarkupsection{Names \label{sec:names}% |
|
478 } |
|
479 \isamarkuptrue% |
|
480 % |
|
481 \begin{isamarkuptext}% |
|
482 In principle, a name is just a string, but there are various |
|
483 convention for encoding additional structure. For example, ``\isa{Foo{\isachardot}bar{\isachardot}baz}'' is considered as a qualified name consisting of |
|
484 three basic name components. The individual constituents of a name |
|
485 may have further substructure, e.g.\ the string |
|
486 ``\verb,\,\verb,<alpha>,'' encodes as a single symbol.% |
|
487 \end{isamarkuptext}% |
|
488 \isamarkuptrue% |
|
489 % |
|
490 \isamarkupsubsection{Strings of symbols% |
|
491 } |
|
492 \isamarkuptrue% |
|
493 % |
|
494 \begin{isamarkuptext}% |
|
495 \glossary{Symbol}{The smallest unit of text in Isabelle, subsumes |
|
496 plain ASCII characters as well as an infinite collection of named |
|
497 symbols (for greek, math etc.).} |
|
498 |
|
499 A \emph{symbol} constitutes the smallest textual unit in Isabelle |
|
500 --- raw characters are normally not encountered at all. Isabelle |
|
501 strings consist of a sequence of symbols, represented as a packed |
|
502 string or a list of strings. Each symbol is in itself a small |
|
503 string, which has either one of the following forms: |
|
504 |
|
505 \begin{enumerate} |
|
506 |
|
507 \item a single ASCII character ``\isa{c}'', for example |
|
508 ``\verb,a,'', |
|
509 |
|
510 \item a regular symbol ``\verb,\,\verb,<,\isa{ident}\verb,>,'', |
|
511 for example ``\verb,\,\verb,<alpha>,'', |
|
512 |
|
513 \item a control symbol ``\verb,\,\verb,<^,\isa{ident}\verb,>,'', |
|
514 for example ``\verb,\,\verb,<^bold>,'', |
|
515 |
|
516 \item a raw symbol ``\verb,\,\verb,<^raw:,\isa{text}\verb,>,'' |
|
517 where \isa{text} constists of printable characters excluding |
|
518 ``\verb,.,'' and ``\verb,>,'', for example |
|
519 ``\verb,\,\verb,<^raw:$\sum_{i = 1}^n$>,'', |
|
520 |
|
521 \item a numbered raw control symbol ``\verb,\,\verb,<^raw,\isa{n}\verb,>, where \isa{n} consists of digits, for example |
|
522 ``\verb,\,\verb,<^raw42>,''. |
|
523 |
|
524 \end{enumerate} |
|
525 |
|
526 \noindent The \isa{ident} syntax for symbol names is \isa{letter\ {\isacharparenleft}letter\ {\isacharbar}\ digit{\isacharparenright}\isactrlsup {\isacharasterisk}}, where \isa{letter\ {\isacharequal}\ A{\isachardot}{\isachardot}Za{\isachardot}{\isachardot}z} and \isa{digit\ {\isacharequal}\ {\isadigit{0}}{\isachardot}{\isachardot}{\isadigit{9}}}. There are infinitely many |
|
527 regular symbols and control symbols, but a fixed collection of |
|
528 standard symbols is treated specifically. For example, |
|
529 ``\verb,\,\verb,<alpha>,'' is classified as a letter, which means it |
|
530 may occur within regular Isabelle identifiers. |
|
531 |
|
532 Since the character set underlying Isabelle symbols is 7-bit ASCII |
|
533 and 8-bit characters are passed through transparently, Isabelle may |
|
534 also process Unicode/UCS data in UTF-8 encoding. Unicode provides |
|
535 its own collection of mathematical symbols, but there is no built-in |
|
536 link to the standard collection of Isabelle. |
|
537 |
|
538 \medskip Output of Isabelle symbols depends on the print mode |
|
539 (\secref{FIXME}). For example, the standard {\LaTeX} setup of the |
|
540 Isabelle document preparation system would present |
|
541 ``\verb,\,\verb,<alpha>,'' as \isa{{\isasymalpha}}, and |
|
542 ``\verb,\,\verb,<^bold>,\verb,\,\verb,<alpha>,'' as \isa{\isactrlbold {\isasymalpha}}.% |
|
543 \end{isamarkuptext}% |
|
544 \isamarkuptrue% |
|
545 % |
|
546 \isadelimmlref |
|
547 % |
|
548 \endisadelimmlref |
|
549 % |
|
550 \isatagmlref |
|
551 % |
|
552 \begin{isamarkuptext}% |
|
553 \begin{mldecls} |
|
554 \indexmltype{Symbol.symbol}\verb|type Symbol.symbol| \\ |
|
555 \indexml{Symbol.explode}\verb|Symbol.explode: string -> Symbol.symbol list| \\ |
|
556 \indexml{Symbol.is\_letter}\verb|Symbol.is_letter: Symbol.symbol -> bool| \\ |
|
557 \indexml{Symbol.is\_digit}\verb|Symbol.is_digit: Symbol.symbol -> bool| \\ |
|
558 \indexml{Symbol.is\_quasi}\verb|Symbol.is_quasi: Symbol.symbol -> bool| \\ |
|
559 \indexml{Symbol.is\_blank}\verb|Symbol.is_blank: Symbol.symbol -> bool| \\ |
|
560 \end{mldecls} |
|
561 \begin{mldecls} |
|
562 \indexmltype{Symbol.sym}\verb|type Symbol.sym| \\ |
|
563 \indexml{Symbol.decode}\verb|Symbol.decode: Symbol.symbol -> Symbol.sym| \\ |
|
564 \end{mldecls} |
|
565 |
|
566 \begin{description} |
|
567 |
|
568 \item \verb|Symbol.symbol| represents individual Isabelle |
|
569 symbols; this is an alias for \verb|string|. |
|
570 |
|
571 \item \verb|Symbol.explode|~\isa{str} produces a symbol list |
|
572 from the packed form. This function supercedes \verb|String.explode| for virtually all purposes of manipulating text in |
|
573 Isabelle! |
|
574 |
|
575 \item \verb|Symbol.is_letter|, \verb|Symbol.is_digit|, \verb|Symbol.is_quasi|, \verb|Symbol.is_blank| classify standard |
|
576 symbols according to fixed syntactic conventions of Isabelle, cf.\ |
|
577 \cite{isabelle-isar-ref}. |
|
578 |
|
579 \item \verb|Symbol.sym| is a concrete datatype that represents |
|
580 the different kinds of symbols explicitly, with constructors \verb|Symbol.Char|, \verb|Symbol.Sym|, \verb|Symbol.Ctrl|, \verb|Symbol.Raw|. |
|
581 |
|
582 \item \verb|Symbol.decode| converts the string representation of a |
|
583 symbol into the datatype version. |
|
584 |
|
585 \end{description}% |
|
586 \end{isamarkuptext}% |
|
587 \isamarkuptrue% |
|
588 % |
|
589 \endisatagmlref |
|
590 {\isafoldmlref}% |
|
591 % |
|
592 \isadelimmlref |
|
593 % |
|
594 \endisadelimmlref |
|
595 % |
|
596 \isamarkupsubsection{Basic names \label{sec:basic-names}% |
|
597 } |
|
598 \isamarkuptrue% |
|
599 % |
|
600 \begin{isamarkuptext}% |
|
601 A \emph{basic name} essentially consists of a single Isabelle |
|
602 identifier. There are conventions to mark separate classes of basic |
|
603 names, by attaching a suffix of underscores (\isa{{\isacharunderscore}}): one |
|
604 underscore means \emph{internal name}, two underscores means |
|
605 \emph{Skolem name}, three underscores means \emph{internal Skolem |
|
606 name}. |
|
607 |
|
608 For example, the basic name \isa{foo} has the internal version |
|
609 \isa{foo{\isacharunderscore}}, with Skolem versions \isa{foo{\isacharunderscore}{\isacharunderscore}} and \isa{foo{\isacharunderscore}{\isacharunderscore}{\isacharunderscore}}, respectively. |
|
610 |
|
611 These special versions provide copies of the basic name space, apart |
|
612 from anything that normally appears in the user text. For example, |
|
613 system generated variables in Isar proof contexts are usually marked |
|
614 as internal, which prevents mysterious name references like \isa{xaa} to appear in the text. |
|
615 |
|
616 \medskip Manipulating binding scopes often requires on-the-fly |
|
617 renamings. A \emph{name context} contains a collection of already |
|
618 used names. The \isa{declare} operation adds names to the |
|
619 context. |
|
620 |
|
621 The \isa{invents} operation derives a number of fresh names from |
|
622 a given starting point. For example, the first three names derived |
|
623 from \isa{a} are \isa{a}, \isa{b}, \isa{c}. |
|
624 |
|
625 The \isa{variants} operation produces fresh names by |
|
626 incrementing tentative names as base-26 numbers (with digits \isa{a{\isachardot}{\isachardot}z}) until all clashes are resolved. For example, name \isa{foo} results in variants \isa{fooa}, \isa{foob}, \isa{fooc}, \dots, \isa{fooaa}, \isa{fooab} etc.; each renaming |
|
627 step picks the next unused variant from this sequence.% |
|
628 \end{isamarkuptext}% |
|
629 \isamarkuptrue% |
|
630 % |
|
631 \isadelimmlref |
|
632 % |
|
633 \endisadelimmlref |
|
634 % |
|
635 \isatagmlref |
|
636 % |
|
637 \begin{isamarkuptext}% |
|
638 \begin{mldecls} |
|
639 \indexml{Name.internal}\verb|Name.internal: string -> string| \\ |
|
640 \indexml{Name.skolem}\verb|Name.skolem: string -> string| \\ |
|
641 \end{mldecls} |
|
642 \begin{mldecls} |
|
643 \indexmltype{Name.context}\verb|type Name.context| \\ |
|
644 \indexml{Name.context}\verb|Name.context: Name.context| \\ |
|
645 \indexml{Name.declare}\verb|Name.declare: string -> Name.context -> Name.context| \\ |
|
646 \indexml{Name.invents}\verb|Name.invents: Name.context -> string -> int -> string list| \\ |
|
647 \indexml{Name.variants}\verb|Name.variants: string list -> Name.context -> string list * Name.context| \\ |
|
648 \end{mldecls} |
|
649 |
|
650 \begin{description} |
|
651 |
|
652 \item \verb|Name.internal|~\isa{name} produces an internal name |
|
653 by adding one underscore. |
|
654 |
|
655 \item \verb|Name.skolem|~\isa{name} produces a Skolem name by |
|
656 adding two underscores. |
|
657 |
|
658 \item \verb|Name.context| represents the context of already used |
|
659 names; the initial value is \verb|Name.context|. |
|
660 |
|
661 \item \verb|Name.declare|~\isa{name} enters a used name into the |
|
662 context. |
|
663 |
|
664 \item \verb|Name.invents|~\isa{context\ name\ n} produces \isa{n} fresh names derived from \isa{name}. |
|
665 |
|
666 \item \verb|Name.variants|~\isa{names\ context} produces fresh |
|
667 varians of \isa{names}; the result is entered into the context. |
|
668 |
|
669 \end{description}% |
|
670 \end{isamarkuptext}% |
|
671 \isamarkuptrue% |
|
672 % |
|
673 \endisatagmlref |
|
674 {\isafoldmlref}% |
|
675 % |
|
676 \isadelimmlref |
|
677 % |
|
678 \endisadelimmlref |
|
679 % |
|
680 \isamarkupsubsection{Indexed names% |
|
681 } |
|
682 \isamarkuptrue% |
|
683 % |
|
684 \begin{isamarkuptext}% |
|
685 An \emph{indexed name} (or \isa{indexname}) is a pair of a basic |
|
686 name and a natural number. This representation allows efficient |
|
687 renaming by incrementing the second component only. The canonical |
|
688 way to rename two collections of indexnames apart from each other is |
|
689 this: determine the maximum index \isa{maxidx} of the first |
|
690 collection, then increment all indexes of the second collection by |
|
691 \isa{maxidx\ {\isacharplus}\ {\isadigit{1}}}; the maximum index of an empty collection is |
|
692 \isa{{\isacharminus}{\isadigit{1}}}. |
|
693 |
|
694 Occasionally, basic names and indexed names are injected into the |
|
695 same pair type: the (improper) indexname \isa{{\isacharparenleft}x{\isacharcomma}\ {\isacharminus}{\isadigit{1}}{\isacharparenright}} is used |
|
696 to encode basic names. |
|
697 |
|
698 \medskip Isabelle syntax observes the following rules for |
|
699 representing an indexname \isa{{\isacharparenleft}x{\isacharcomma}\ i{\isacharparenright}} as a packed string: |
|
700 |
|
701 \begin{itemize} |
|
702 |
|
703 \item \isa{{\isacharquery}x} if \isa{x} does not end with a digit and \isa{i\ {\isacharequal}\ {\isadigit{0}}}, |
|
704 |
|
705 \item \isa{{\isacharquery}xi} if \isa{x} does not end with a digit, |
|
706 |
|
707 \item \isa{{\isacharquery}x{\isachardot}i} otherwise. |
|
708 |
|
709 \end{itemize} |
|
710 |
|
711 Indexnames may acquire large index numbers over time. Results are |
|
712 normalized towards \isa{{\isadigit{0}}} at certain checkpoints, notably at |
|
713 the end of a proof. This works by producing variants of the |
|
714 corresponding basic name components. For example, the collection |
|
715 \isa{{\isacharquery}x{\isadigit{1}}{\isacharcomma}\ {\isacharquery}x{\isadigit{7}}{\isacharcomma}\ {\isacharquery}x{\isadigit{4}}{\isadigit{2}}} becomes \isa{{\isacharquery}x{\isacharcomma}\ {\isacharquery}xa{\isacharcomma}\ {\isacharquery}xb}.% |
|
716 \end{isamarkuptext}% |
|
717 \isamarkuptrue% |
|
718 % |
|
719 \isadelimmlref |
|
720 % |
|
721 \endisadelimmlref |
|
722 % |
|
723 \isatagmlref |
|
724 % |
|
725 \begin{isamarkuptext}% |
|
726 \begin{mldecls} |
|
727 \indexmltype{indexname}\verb|type indexname| \\ |
|
728 \end{mldecls} |
|
729 |
|
730 \begin{description} |
|
731 |
|
732 \item \verb|indexname| represents indexed names. This is an |
|
733 abbreviation for \verb|string * int|. The second component is |
|
734 usually non-negative, except for situations where \isa{{\isacharparenleft}x{\isacharcomma}\ {\isacharminus}{\isadigit{1}}{\isacharparenright}} |
|
735 is used to embed basic names into this type. |
|
736 |
|
737 \end{description}% |
|
738 \end{isamarkuptext}% |
|
739 \isamarkuptrue% |
|
740 % |
|
741 \endisatagmlref |
|
742 {\isafoldmlref}% |
|
743 % |
|
744 \isadelimmlref |
|
745 % |
|
746 \endisadelimmlref |
|
747 % |
|
748 \isamarkupsubsection{Qualified names and name spaces% |
|
749 } |
|
750 \isamarkuptrue% |
|
751 % |
|
752 \begin{isamarkuptext}% |
|
753 A \emph{qualified name} consists of a non-empty sequence of basic |
|
754 name components. The packed representation uses a dot as separator, |
|
755 as in ``\isa{A{\isachardot}b{\isachardot}c}''. The last component is called \emph{base} |
|
756 name, the remaining prefix \emph{qualifier} (which may be empty). |
|
757 The idea of qualified names is to encode nested structures by |
|
758 recording the access paths as qualifiers. For example, an item |
|
759 named ``\isa{A{\isachardot}b{\isachardot}c}'' may be understood as a local entity \isa{c}, within a local structure \isa{b}, within a global |
|
760 structure \isa{A}. Typically, name space hierarchies consist of |
|
761 1--2 levels of qualification, but this need not be always so. |
|
762 |
|
763 The empty name is commonly used as an indication of unnamed |
|
764 entities, whenever this makes any sense. The basic operations on |
|
765 qualified names are smart enough to pass through such improper names |
|
766 unchanged. |
|
767 |
|
768 \medskip A \isa{naming} policy tells how to turn a name |
|
769 specification into a fully qualified internal name (by the \isa{full} operation), and how fully qualified names may be accessed |
|
770 externally. For example, the default naming policy is to prefix an |
|
771 implicit path: \isa{full\ x} produces \isa{path{\isachardot}x}, and the |
|
772 standard accesses for \isa{path{\isachardot}x} include both \isa{x} and |
|
773 \isa{path{\isachardot}x}. Normally, the naming is implicit in the theory or |
|
774 proof context; there are separate versions of the corresponding. |
|
775 |
|
776 \medskip A \isa{name\ space} manages a collection of fully |
|
777 internalized names, together with a mapping between external names |
|
778 and internal names (in both directions). The corresponding \isa{intern} and \isa{extern} operations are mostly used for |
|
779 parsing and printing only! The \isa{declare} operation augments |
|
780 a name space according to the accesses determined by the naming |
|
781 policy. |
|
782 |
|
783 \medskip As a general principle, there is a separate name space for |
|
784 each kind of formal entity, e.g.\ logical constant, type |
|
785 constructor, type class, theorem. It is usually clear from the |
|
786 occurrence in concrete syntax (or from the scope) which kind of |
|
787 entity a name refers to. For example, the very same name \isa{c} may be used uniformly for a constant, type constructor, and |
|
788 type class. |
|
789 |
|
790 There are common schemes to name theorems systematically, according |
|
791 to the name of the main logical entity involved, e.g.\ \isa{c{\isachardot}intro} for a canonical theorem related to constant \isa{c}. |
|
792 This technique of mapping names from one space into another requires |
|
793 some care in order to avoid conflicts. In particular, theorem names |
|
794 derived from a type constructor or type class are better suffixed in |
|
795 addition to the usual qualification, e.g.\ \isa{c{\isacharunderscore}type{\isachardot}intro} |
|
796 and \isa{c{\isacharunderscore}class{\isachardot}intro} for theorems related to type \isa{c} |
|
797 and class \isa{c}, respectively.% |
|
798 \end{isamarkuptext}% |
|
799 \isamarkuptrue% |
|
800 % |
|
801 \isadelimmlref |
|
802 % |
|
803 \endisadelimmlref |
|
804 % |
|
805 \isatagmlref |
|
806 % |
|
807 \begin{isamarkuptext}% |
|
808 \begin{mldecls} |
|
809 \indexml{NameSpace.base}\verb|NameSpace.base: string -> string| \\ |
|
810 \indexml{NameSpace.qualifier}\verb|NameSpace.qualifier: string -> string| \\ |
|
811 \indexml{NameSpace.append}\verb|NameSpace.append: string -> string -> string| \\ |
|
812 \indexml{NameSpace.implode}\verb|NameSpace.implode: string list -> string| \\ |
|
813 \indexml{NameSpace.explode}\verb|NameSpace.explode: string -> string list| \\ |
|
814 \end{mldecls} |
|
815 \begin{mldecls} |
|
816 \indexmltype{NameSpace.naming}\verb|type NameSpace.naming| \\ |
|
817 \indexml{NameSpace.default\_naming}\verb|NameSpace.default_naming: NameSpace.naming| \\ |
|
818 \indexml{NameSpace.add\_path}\verb|NameSpace.add_path: string -> NameSpace.naming -> NameSpace.naming| \\ |
|
819 \indexml{NameSpace.full\_name}\verb|NameSpace.full_name: NameSpace.naming -> binding -> string| \\ |
|
820 \end{mldecls} |
|
821 \begin{mldecls} |
|
822 \indexmltype{NameSpace.T}\verb|type NameSpace.T| \\ |
|
823 \indexml{NameSpace.empty}\verb|NameSpace.empty: NameSpace.T| \\ |
|
824 \indexml{NameSpace.merge}\verb|NameSpace.merge: NameSpace.T * NameSpace.T -> NameSpace.T| \\ |
|
825 \indexml{NameSpace.declare}\verb|NameSpace.declare: NameSpace.naming -> binding -> NameSpace.T -> string * NameSpace.T| \\ |
|
826 \indexml{NameSpace.intern}\verb|NameSpace.intern: NameSpace.T -> string -> string| \\ |
|
827 \indexml{NameSpace.extern}\verb|NameSpace.extern: NameSpace.T -> string -> string| \\ |
|
828 \end{mldecls} |
|
829 |
|
830 \begin{description} |
|
831 |
|
832 \item \verb|NameSpace.base|~\isa{name} returns the base name of a |
|
833 qualified name. |
|
834 |
|
835 \item \verb|NameSpace.qualifier|~\isa{name} returns the qualifier |
|
836 of a qualified name. |
|
837 |
|
838 \item \verb|NameSpace.append|~\isa{name\isactrlisub {\isadigit{1}}\ name\isactrlisub {\isadigit{2}}} |
|
839 appends two qualified names. |
|
840 |
|
841 \item \verb|NameSpace.implode|~\isa{name} and \verb|NameSpace.explode|~\isa{names} convert between the packed string |
|
842 representation and the explicit list form of qualified names. |
|
843 |
|
844 \item \verb|NameSpace.naming| represents the abstract concept of |
|
845 a naming policy. |
|
846 |
|
847 \item \verb|NameSpace.default_naming| is the default naming policy. |
|
848 In a theory context, this is usually augmented by a path prefix |
|
849 consisting of the theory name. |
|
850 |
|
851 \item \verb|NameSpace.add_path|~\isa{path\ naming} augments the |
|
852 naming policy by extending its path component. |
|
853 |
|
854 \item \verb|NameSpace.full_name|\isa{naming\ binding} turns a name |
|
855 binding (usually a basic name) into the fully qualified |
|
856 internal name, according to the given naming policy. |
|
857 |
|
858 \item \verb|NameSpace.T| represents name spaces. |
|
859 |
|
860 \item \verb|NameSpace.empty| and \verb|NameSpace.merge|~\isa{{\isacharparenleft}space\isactrlisub {\isadigit{1}}{\isacharcomma}\ space\isactrlisub {\isadigit{2}}{\isacharparenright}} are the canonical operations for |
|
861 maintaining name spaces according to theory data management |
|
862 (\secref{sec:context-data}). |
|
863 |
|
864 \item \verb|NameSpace.declare|~\isa{naming\ bindings\ space} enters a |
|
865 name binding as fully qualified internal name into the name space, |
|
866 with external accesses determined by the naming policy. |
|
867 |
|
868 \item \verb|NameSpace.intern|~\isa{space\ name} internalizes a |
|
869 (partially qualified) external name. |
|
870 |
|
871 This operation is mostly for parsing! Note that fully qualified |
|
872 names stemming from declarations are produced via \verb|NameSpace.full_name| and \verb|NameSpace.declare| |
|
873 (or their derivatives for \verb|theory| and |
|
874 \verb|Proof.context|). |
|
875 |
|
876 \item \verb|NameSpace.extern|~\isa{space\ name} externalizes a |
|
877 (fully qualified) internal name. |
|
878 |
|
879 This operation is mostly for printing! Note unqualified names are |
|
880 produced via \verb|NameSpace.base|. |
|
881 |
|
882 \end{description}% |
|
883 \end{isamarkuptext}% |
|
884 \isamarkuptrue% |
|
885 % |
|
886 \endisatagmlref |
|
887 {\isafoldmlref}% |
|
888 % |
|
889 \isadelimmlref |
|
890 % |
|
891 \endisadelimmlref |
|
892 % |
|
893 \isadelimtheory |
|
894 % |
|
895 \endisadelimtheory |
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896 % |
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897 \isatagtheory |
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898 \isacommand{end}\isamarkupfalse% |
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899 % |
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900 \endisatagtheory |
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901 {\isafoldtheory}% |
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902 % |
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903 \isadelimtheory |
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904 % |
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905 \endisadelimtheory |
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906 \isanewline |
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907 \end{isabellebody}% |
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908 %%% Local Variables: |
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909 %%% mode: latex |
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910 %%% TeX-master: "root" |
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911 %%% End: |
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