1 theory Prelim |
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2 imports Base |
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3 begin |
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4 |
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5 chapter {* Preliminaries *} |
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6 |
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7 section {* Contexts \label{sec:context} *} |
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8 |
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9 text {* |
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10 A logical context represents the background that is required for |
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11 formulating statements and composing proofs. It acts as a medium to |
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12 produce formal content, depending on earlier material (declarations, |
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13 results etc.). |
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14 |
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15 For example, derivations within the Isabelle/Pure logic can be |
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16 described as a judgment @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<phi>"}, which means that a |
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17 proposition @{text "\<phi>"} is derivable from hypotheses @{text "\<Gamma>"} |
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18 within the theory @{text "\<Theta>"}. There are logical reasons for |
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19 keeping @{text "\<Theta>"} and @{text "\<Gamma>"} separate: theories can be |
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20 liberal about supporting type constructors and schematic |
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21 polymorphism of constants and axioms, while the inner calculus of |
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22 @{text "\<Gamma> \<turnstile> \<phi>"} is strictly limited to Simple Type Theory (with |
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23 fixed type variables in the assumptions). |
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24 |
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25 \medskip Contexts and derivations are linked by the following key |
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26 principles: |
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27 |
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28 \begin{itemize} |
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29 |
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30 \item Transfer: monotonicity of derivations admits results to be |
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31 transferred into a \emph{larger} context, i.e.\ @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> |
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32 \<phi>"} implies @{text "\<Gamma>' \<turnstile>\<^sub>\<Theta>\<^sub>' \<phi>"} for contexts @{text "\<Theta>' |
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33 \<supseteq> \<Theta>"} and @{text "\<Gamma>' \<supseteq> \<Gamma>"}. |
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34 |
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35 \item Export: discharge of hypotheses admits results to be exported |
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36 into a \emph{smaller} context, i.e.\ @{text "\<Gamma>' \<turnstile>\<^sub>\<Theta> \<phi>"} |
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37 implies @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<Delta> \<Longrightarrow> \<phi>"} where @{text "\<Gamma>' \<supseteq> \<Gamma>"} and |
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38 @{text "\<Delta> = \<Gamma>' - \<Gamma>"}. Note that @{text "\<Theta>"} remains unchanged here, |
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39 only the @{text "\<Gamma>"} part is affected. |
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40 |
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41 \end{itemize} |
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42 |
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43 \medskip By modeling the main characteristics of the primitive |
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44 @{text "\<Theta>"} and @{text "\<Gamma>"} above, and abstracting over any |
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45 particular logical content, we arrive at the fundamental notions of |
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46 \emph{theory context} and \emph{proof context} in Isabelle/Isar. |
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47 These implement a certain policy to manage arbitrary \emph{context |
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48 data}. There is a strongly-typed mechanism to declare new kinds of |
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49 data at compile time. |
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50 |
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51 The internal bootstrap process of Isabelle/Pure eventually reaches a |
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52 stage where certain data slots provide the logical content of @{text |
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53 "\<Theta>"} and @{text "\<Gamma>"} sketched above, but this does not stop there! |
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54 Various additional data slots support all kinds of mechanisms that |
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55 are not necessarily part of the core logic. |
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56 |
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57 For example, there would be data for canonical introduction and |
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58 elimination rules for arbitrary operators (depending on the |
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59 object-logic and application), which enables users to perform |
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60 standard proof steps implicitly (cf.\ the @{text "rule"} method |
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61 \cite{isabelle-isar-ref}). |
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62 |
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63 \medskip Thus Isabelle/Isar is able to bring forth more and more |
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64 concepts successively. In particular, an object-logic like |
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65 Isabelle/HOL continues the Isabelle/Pure setup by adding specific |
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66 components for automated reasoning (classical reasoner, tableau |
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67 prover, structured induction etc.) and derived specification |
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68 mechanisms (inductive predicates, recursive functions etc.). All of |
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69 this is ultimately based on the generic data management by theory |
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70 and proof contexts introduced here. |
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71 *} |
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72 |
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73 |
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74 subsection {* Theory context \label{sec:context-theory} *} |
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75 |
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76 text {* A \emph{theory} is a data container with explicit name and |
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77 unique identifier. Theories are related by a (nominal) sub-theory |
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78 relation, which corresponds to the dependency graph of the original |
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79 construction; each theory is derived from a certain sub-graph of |
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80 ancestor theories. To this end, the system maintains a set of |
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81 symbolic ``identification stamps'' within each theory. |
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82 |
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83 The @{text "merge"} operation produces the least upper bound of two |
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84 theories, which actually degenerates into absorption of one theory |
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85 into the other (according to the nominal sub-theory relation). |
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86 |
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87 The @{text "begin"} operation starts a new theory by importing |
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88 several parent theories and entering a special mode of nameless |
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89 incremental updates, until the final @{text "end"} operation is |
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90 performed. |
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91 |
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92 \medskip The example in \figref{fig:ex-theory} below shows a theory |
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93 graph derived from @{text "Pure"}, with theory @{text "Length"} |
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94 importing @{text "Nat"} and @{text "List"}. The body of @{text |
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95 "Length"} consists of a sequence of updates, resulting in locally a |
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96 linear sub-theory relation for each intermediate step. |
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97 |
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98 \begin{figure}[htb] |
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99 \begin{center} |
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100 \begin{tabular}{rcccl} |
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101 & & @{text "Pure"} \\ |
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102 & & @{text "\<down>"} \\ |
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103 & & @{text "FOL"} \\ |
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104 & $\swarrow$ & & $\searrow$ & \\ |
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105 @{text "Nat"} & & & & @{text "List"} \\ |
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106 & $\searrow$ & & $\swarrow$ \\ |
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107 & & @{text "Length"} \\ |
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108 & & \multicolumn{3}{l}{~~@{keyword "begin"}} \\ |
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109 & & $\vdots$~~ \\ |
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110 & & \multicolumn{3}{l}{~~@{command "end"}} \\ |
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111 \end{tabular} |
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112 \caption{A theory definition depending on ancestors}\label{fig:ex-theory} |
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113 \end{center} |
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114 \end{figure} |
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115 |
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116 \medskip Derived formal entities may retain a reference to the |
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117 background theory in order to indicate the formal context from which |
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118 they were produced. This provides an immutable certificate of the |
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119 background theory. *} |
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120 |
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121 text %mlref {* |
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122 \begin{mldecls} |
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123 @{index_ML_type theory} \\ |
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124 @{index_ML Theory.eq_thy: "theory * theory -> bool"} \\ |
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125 @{index_ML Theory.subthy: "theory * theory -> bool"} \\ |
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126 @{index_ML Theory.merge: "theory * theory -> theory"} \\ |
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127 @{index_ML Theory.begin_theory: "string * Position.T -> theory list -> theory"} \\ |
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128 @{index_ML Theory.parents_of: "theory -> theory list"} \\ |
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129 @{index_ML Theory.ancestors_of: "theory -> theory list"} \\ |
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130 \end{mldecls} |
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131 |
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132 \begin{description} |
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133 |
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134 \item Type @{ML_type theory} represents theory contexts. |
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135 |
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136 \item @{ML "Theory.eq_thy"}~@{text "(thy\<^sub>1, thy\<^sub>2)"} check strict |
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137 identity of two theories. |
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138 |
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139 \item @{ML "Theory.subthy"}~@{text "(thy\<^sub>1, thy\<^sub>2)"} compares theories |
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140 according to the intrinsic graph structure of the construction. |
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141 This sub-theory relation is a nominal approximation of inclusion |
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142 (@{text "\<subseteq>"}) of the corresponding content (according to the |
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143 semantics of the ML modules that implement the data). |
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144 |
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145 \item @{ML "Theory.merge"}~@{text "(thy\<^sub>1, thy\<^sub>2)"} absorbs one theory |
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146 into the other. This version of ad-hoc theory merge fails for |
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147 unrelated theories! |
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148 |
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149 \item @{ML "Theory.begin_theory"}~@{text "name parents"} constructs |
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150 a new theory based on the given parents. This ML function is |
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151 normally not invoked directly. |
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152 |
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153 \item @{ML "Theory.parents_of"}~@{text "thy"} returns the direct |
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154 ancestors of @{text thy}. |
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155 |
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156 \item @{ML "Theory.ancestors_of"}~@{text "thy"} returns all |
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157 ancestors of @{text thy} (not including @{text thy} itself). |
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158 |
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159 \end{description} |
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160 *} |
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161 |
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162 text %mlantiq {* |
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163 \begin{matharray}{rcl} |
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164 @{ML_antiquotation_def "theory"} & : & @{text ML_antiquotation} \\ |
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165 @{ML_antiquotation_def "theory_context"} & : & @{text ML_antiquotation} \\ |
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166 \end{matharray} |
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167 |
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168 @{rail \<open> |
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169 @@{ML_antiquotation theory} nameref? |
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170 ; |
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171 @@{ML_antiquotation theory_context} nameref |
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172 \<close>} |
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173 |
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174 \begin{description} |
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175 |
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176 \item @{text "@{theory}"} refers to the background theory of the |
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177 current context --- as abstract value. |
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178 |
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179 \item @{text "@{theory A}"} refers to an explicitly named ancestor |
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180 theory @{text "A"} of the background theory of the current context |
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181 --- as abstract value. |
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182 |
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183 \item @{text "@{theory_context A}"} is similar to @{text "@{theory |
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184 A}"}, but presents the result as initial @{ML_type Proof.context} |
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185 (see also @{ML Proof_Context.init_global}). |
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186 |
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187 \end{description} |
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188 *} |
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189 |
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190 |
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191 subsection {* Proof context \label{sec:context-proof} *} |
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192 |
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193 text {* A proof context is a container for pure data that refers to |
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194 the theory from which it is derived. The @{text "init"} operation |
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195 creates a proof context from a given theory. There is an explicit |
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196 @{text "transfer"} operation to force resynchronization with updates |
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197 to the background theory -- this is rarely required in practice. |
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198 |
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199 Entities derived in a proof context need to record logical |
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200 requirements explicitly, since there is no separate context |
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201 identification or symbolic inclusion as for theories. For example, |
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202 hypotheses used in primitive derivations (cf.\ \secref{sec:thms}) |
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203 are recorded separately within the sequent @{text "\<Gamma> \<turnstile> \<phi>"}, just to |
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204 make double sure. Results could still leak into an alien proof |
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205 context due to programming errors, but Isabelle/Isar includes some |
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206 extra validity checks in critical positions, notably at the end of a |
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207 sub-proof. |
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208 |
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209 Proof contexts may be manipulated arbitrarily, although the common |
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210 discipline is to follow block structure as a mental model: a given |
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211 context is extended consecutively, and results are exported back |
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212 into the original context. Note that an Isar proof state models |
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213 block-structured reasoning explicitly, using a stack of proof |
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214 contexts internally. For various technical reasons, the background |
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215 theory of an Isar proof state must not be changed while the proof is |
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216 still under construction! |
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217 *} |
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218 |
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219 text %mlref {* |
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220 \begin{mldecls} |
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221 @{index_ML_type Proof.context} \\ |
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222 @{index_ML Proof_Context.init_global: "theory -> Proof.context"} \\ |
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223 @{index_ML Proof_Context.theory_of: "Proof.context -> theory"} \\ |
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224 @{index_ML Proof_Context.transfer: "theory -> Proof.context -> Proof.context"} \\ |
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225 \end{mldecls} |
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226 |
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227 \begin{description} |
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228 |
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229 \item Type @{ML_type Proof.context} represents proof contexts. |
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230 |
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231 \item @{ML Proof_Context.init_global}~@{text "thy"} produces a proof |
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232 context derived from @{text "thy"}, initializing all data. |
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233 |
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234 \item @{ML Proof_Context.theory_of}~@{text "ctxt"} selects the |
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235 background theory from @{text "ctxt"}. |
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236 |
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237 \item @{ML Proof_Context.transfer}~@{text "thy ctxt"} promotes the |
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238 background theory of @{text "ctxt"} to the super theory @{text |
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239 "thy"}. |
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240 |
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241 \end{description} |
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242 *} |
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243 |
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244 text %mlantiq {* |
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245 \begin{matharray}{rcl} |
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246 @{ML_antiquotation_def "context"} & : & @{text ML_antiquotation} \\ |
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247 \end{matharray} |
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248 |
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249 \begin{description} |
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250 |
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251 \item @{text "@{context}"} refers to \emph{the} context at |
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252 compile-time --- as abstract value. Independently of (local) theory |
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253 or proof mode, this always produces a meaningful result. |
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254 |
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255 This is probably the most common antiquotation in interactive |
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256 experimentation with ML inside Isar. |
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257 |
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258 \end{description} |
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259 *} |
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260 |
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261 |
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262 subsection {* Generic contexts \label{sec:generic-context} *} |
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263 |
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264 text {* |
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265 A generic context is the disjoint sum of either a theory or proof |
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266 context. Occasionally, this enables uniform treatment of generic |
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267 context data, typically extra-logical information. Operations on |
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268 generic contexts include the usual injections, partial selections, |
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269 and combinators for lifting operations on either component of the |
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270 disjoint sum. |
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271 |
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272 Moreover, there are total operations @{text "theory_of"} and @{text |
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273 "proof_of"} to convert a generic context into either kind: a theory |
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274 can always be selected from the sum, while a proof context might |
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275 have to be constructed by an ad-hoc @{text "init"} operation, which |
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276 incurs a small runtime overhead. |
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277 *} |
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278 |
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279 text %mlref {* |
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280 \begin{mldecls} |
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281 @{index_ML_type Context.generic} \\ |
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282 @{index_ML Context.theory_of: "Context.generic -> theory"} \\ |
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283 @{index_ML Context.proof_of: "Context.generic -> Proof.context"} \\ |
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284 \end{mldecls} |
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285 |
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286 \begin{description} |
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287 |
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288 \item Type @{ML_type Context.generic} is the direct sum of @{ML_type |
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289 "theory"} and @{ML_type "Proof.context"}, with the datatype |
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290 constructors @{ML "Context.Theory"} and @{ML "Context.Proof"}. |
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291 |
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292 \item @{ML Context.theory_of}~@{text "context"} always produces a |
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293 theory from the generic @{text "context"}, using @{ML |
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294 "Proof_Context.theory_of"} as required. |
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295 |
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296 \item @{ML Context.proof_of}~@{text "context"} always produces a |
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297 proof context from the generic @{text "context"}, using @{ML |
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298 "Proof_Context.init_global"} as required (note that this re-initializes the |
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299 context data with each invocation). |
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300 |
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301 \end{description} |
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302 *} |
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303 |
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304 |
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305 subsection {* Context data \label{sec:context-data} *} |
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306 |
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307 text {* The main purpose of theory and proof contexts is to manage |
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308 arbitrary (pure) data. New data types can be declared incrementally |
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309 at compile time. There are separate declaration mechanisms for any |
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310 of the three kinds of contexts: theory, proof, generic. |
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311 |
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312 \paragraph{Theory data} declarations need to implement the following |
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313 SML signature: |
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314 |
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315 \medskip |
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316 \begin{tabular}{ll} |
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317 @{text "\<type> T"} & representing type \\ |
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318 @{text "\<val> empty: T"} & empty default value \\ |
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319 @{text "\<val> extend: T \<rightarrow> T"} & re-initialize on import \\ |
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320 @{text "\<val> merge: T \<times> T \<rightarrow> T"} & join on import \\ |
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321 \end{tabular} |
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322 \medskip |
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323 |
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324 The @{text "empty"} value acts as initial default for \emph{any} |
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325 theory that does not declare actual data content; @{text "extend"} |
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326 is acts like a unitary version of @{text "merge"}. |
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327 |
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328 Implementing @{text "merge"} can be tricky. The general idea is |
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329 that @{text "merge (data\<^sub>1, data\<^sub>2)"} inserts those parts of @{text |
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330 "data\<^sub>2"} into @{text "data\<^sub>1"} that are not yet present, while |
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331 keeping the general order of things. The @{ML Library.merge} |
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332 function on plain lists may serve as canonical template. |
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333 |
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334 Particularly note that shared parts of the data must not be |
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335 duplicated by naive concatenation, or a theory graph that is like a |
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336 chain of diamonds would cause an exponential blowup! |
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337 |
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338 \paragraph{Proof context data} declarations need to implement the |
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339 following SML signature: |
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340 |
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341 \medskip |
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342 \begin{tabular}{ll} |
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343 @{text "\<type> T"} & representing type \\ |
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344 @{text "\<val> init: theory \<rightarrow> T"} & produce initial value \\ |
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345 \end{tabular} |
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346 \medskip |
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347 |
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348 The @{text "init"} operation is supposed to produce a pure value |
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349 from the given background theory and should be somehow |
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350 ``immediate''. Whenever a proof context is initialized, which |
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351 happens frequently, the the system invokes the @{text "init"} |
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352 operation of \emph{all} theory data slots ever declared. This also |
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353 means that one needs to be economic about the total number of proof |
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354 data declarations in the system, i.e.\ each ML module should declare |
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355 at most one, sometimes two data slots for its internal use. |
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356 Repeated data declarations to simulate a record type should be |
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357 avoided! |
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358 |
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359 \paragraph{Generic data} provides a hybrid interface for both theory |
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360 and proof data. The @{text "init"} operation for proof contexts is |
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361 predefined to select the current data value from the background |
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362 theory. |
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363 |
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364 \bigskip Any of the above data declarations over type @{text "T"} |
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365 result in an ML structure with the following signature: |
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366 |
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367 \medskip |
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368 \begin{tabular}{ll} |
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369 @{text "get: context \<rightarrow> T"} \\ |
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370 @{text "put: T \<rightarrow> context \<rightarrow> context"} \\ |
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371 @{text "map: (T \<rightarrow> T) \<rightarrow> context \<rightarrow> context"} \\ |
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372 \end{tabular} |
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373 \medskip |
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374 |
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375 These other operations provide exclusive access for the particular |
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376 kind of context (theory, proof, or generic context). This interface |
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377 observes the ML discipline for types and scopes: there is no other |
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378 way to access the corresponding data slot of a context. By keeping |
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379 these operations private, an Isabelle/ML module may maintain |
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380 abstract values authentically. *} |
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381 |
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382 text %mlref {* |
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383 \begin{mldecls} |
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384 @{index_ML_functor Theory_Data} \\ |
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385 @{index_ML_functor Proof_Data} \\ |
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386 @{index_ML_functor Generic_Data} \\ |
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387 \end{mldecls} |
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388 |
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389 \begin{description} |
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390 |
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391 \item @{ML_functor Theory_Data}@{text "(spec)"} declares data for |
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392 type @{ML_type theory} according to the specification provided as |
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393 argument structure. The resulting structure provides data init and |
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394 access operations as described above. |
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395 |
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396 \item @{ML_functor Proof_Data}@{text "(spec)"} is analogous to |
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397 @{ML_functor Theory_Data} for type @{ML_type Proof.context}. |
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398 |
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399 \item @{ML_functor Generic_Data}@{text "(spec)"} is analogous to |
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400 @{ML_functor Theory_Data} for type @{ML_type Context.generic}. |
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401 |
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402 \end{description} |
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403 *} |
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404 |
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405 text %mlex {* |
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406 The following artificial example demonstrates theory |
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407 data: we maintain a set of terms that are supposed to be wellformed |
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408 wrt.\ the enclosing theory. The public interface is as follows: |
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409 *} |
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410 |
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411 ML {* |
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412 signature WELLFORMED_TERMS = |
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413 sig |
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414 val get: theory -> term list |
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415 val add: term -> theory -> theory |
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416 end; |
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417 *} |
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418 |
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419 text {* The implementation uses private theory data internally, and |
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420 only exposes an operation that involves explicit argument checking |
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421 wrt.\ the given theory. *} |
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422 |
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423 ML {* |
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424 structure Wellformed_Terms: WELLFORMED_TERMS = |
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425 struct |
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426 |
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427 structure Terms = Theory_Data |
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428 ( |
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429 type T = term Ord_List.T; |
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430 val empty = []; |
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431 val extend = I; |
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432 fun merge (ts1, ts2) = |
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433 Ord_List.union Term_Ord.fast_term_ord ts1 ts2; |
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434 ); |
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435 |
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436 val get = Terms.get; |
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437 |
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438 fun add raw_t thy = |
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439 let |
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440 val t = Sign.cert_term thy raw_t; |
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441 in |
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442 Terms.map (Ord_List.insert Term_Ord.fast_term_ord t) thy |
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443 end; |
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444 |
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445 end; |
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446 *} |
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447 |
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448 text {* Type @{ML_type "term Ord_List.T"} is used for reasonably |
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449 efficient representation of a set of terms: all operations are |
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450 linear in the number of stored elements. Here we assume that users |
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451 of this module do not care about the declaration order, since that |
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452 data structure forces its own arrangement of elements. |
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453 |
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454 Observe how the @{ML_text merge} operation joins the data slots of |
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455 the two constituents: @{ML Ord_List.union} prevents duplication of |
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456 common data from different branches, thus avoiding the danger of |
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457 exponential blowup. Plain list append etc.\ must never be used for |
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458 theory data merges! |
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459 |
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460 \medskip Our intended invariant is achieved as follows: |
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461 \begin{enumerate} |
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462 |
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463 \item @{ML Wellformed_Terms.add} only admits terms that have passed |
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464 the @{ML Sign.cert_term} check of the given theory at that point. |
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465 |
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466 \item Wellformedness in the sense of @{ML Sign.cert_term} is |
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467 monotonic wrt.\ the sub-theory relation. So our data can move |
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468 upwards in the hierarchy (via extension or merges), and maintain |
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469 wellformedness without further checks. |
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470 |
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471 \end{enumerate} |
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472 |
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473 Note that all basic operations of the inference kernel (which |
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474 includes @{ML Sign.cert_term}) observe this monotonicity principle, |
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475 but other user-space tools don't. For example, fully-featured |
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476 type-inference via @{ML Syntax.check_term} (cf.\ |
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477 \secref{sec:term-check}) is not necessarily monotonic wrt.\ the |
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478 background theory, since constraints of term constants can be |
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479 modified by later declarations, for example. |
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480 |
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481 In most cases, user-space context data does not have to take such |
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482 invariants too seriously. The situation is different in the |
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483 implementation of the inference kernel itself, which uses the very |
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484 same data mechanisms for types, constants, axioms etc. |
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485 *} |
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486 |
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487 |
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488 subsection {* Configuration options \label{sec:config-options} *} |
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489 |
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490 text {* A \emph{configuration option} is a named optional value of |
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491 some basic type (Boolean, integer, string) that is stored in the |
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492 context. It is a simple application of general context data |
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493 (\secref{sec:context-data}) that is sufficiently common to justify |
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494 customized setup, which includes some concrete declarations for |
|
495 end-users using existing notation for attributes (cf.\ |
|
496 \secref{sec:attributes}). |
|
497 |
|
498 For example, the predefined configuration option @{attribute |
|
499 show_types} controls output of explicit type constraints for |
|
500 variables in printed terms (cf.\ \secref{sec:read-print}). Its |
|
501 value can be modified within Isar text like this: |
|
502 *} |
|
503 |
|
504 declare [[show_types = false]] |
|
505 -- {* declaration within (local) theory context *} |
|
506 |
|
507 notepad |
|
508 begin |
|
509 note [[show_types = true]] |
|
510 -- {* declaration within proof (forward mode) *} |
|
511 term x |
|
512 |
|
513 have "x = x" |
|
514 using [[show_types = false]] |
|
515 -- {* declaration within proof (backward mode) *} |
|
516 .. |
|
517 end |
|
518 |
|
519 text {* Configuration options that are not set explicitly hold a |
|
520 default value that can depend on the application context. This |
|
521 allows to retrieve the value from another slot within the context, |
|
522 or fall back on a global preference mechanism, for example. |
|
523 |
|
524 The operations to declare configuration options and get/map their |
|
525 values are modeled as direct replacements for historic global |
|
526 references, only that the context is made explicit. This allows |
|
527 easy configuration of tools, without relying on the execution order |
|
528 as required for old-style mutable references. *} |
|
529 |
|
530 text %mlref {* |
|
531 \begin{mldecls} |
|
532 @{index_ML Config.get: "Proof.context -> 'a Config.T -> 'a"} \\ |
|
533 @{index_ML Config.map: "'a Config.T -> ('a -> 'a) -> Proof.context -> Proof.context"} \\ |
|
534 @{index_ML Attrib.setup_config_bool: "binding -> (Context.generic -> bool) -> |
|
535 bool Config.T"} \\ |
|
536 @{index_ML Attrib.setup_config_int: "binding -> (Context.generic -> int) -> |
|
537 int Config.T"} \\ |
|
538 @{index_ML Attrib.setup_config_real: "binding -> (Context.generic -> real) -> |
|
539 real Config.T"} \\ |
|
540 @{index_ML Attrib.setup_config_string: "binding -> (Context.generic -> string) -> |
|
541 string Config.T"} \\ |
|
542 \end{mldecls} |
|
543 |
|
544 \begin{description} |
|
545 |
|
546 \item @{ML Config.get}~@{text "ctxt config"} gets the value of |
|
547 @{text "config"} in the given context. |
|
548 |
|
549 \item @{ML Config.map}~@{text "config f ctxt"} updates the context |
|
550 by updating the value of @{text "config"}. |
|
551 |
|
552 \item @{text "config ="}~@{ML Attrib.setup_config_bool}~@{text "name |
|
553 default"} creates a named configuration option of type @{ML_type |
|
554 bool}, with the given @{text "default"} depending on the application |
|
555 context. The resulting @{text "config"} can be used to get/map its |
|
556 value in a given context. There is an implicit update of the |
|
557 background theory that registers the option as attribute with some |
|
558 concrete syntax. |
|
559 |
|
560 \item @{ML Attrib.config_int}, @{ML Attrib.config_real}, and @{ML |
|
561 Attrib.config_string} work like @{ML Attrib.config_bool}, but for |
|
562 types @{ML_type int} and @{ML_type string}, respectively. |
|
563 |
|
564 \end{description} |
|
565 *} |
|
566 |
|
567 text %mlex {* The following example shows how to declare and use a |
|
568 Boolean configuration option called @{text "my_flag"} with constant |
|
569 default value @{ML false}. *} |
|
570 |
|
571 ML {* |
|
572 val my_flag = |
|
573 Attrib.setup_config_bool @{binding my_flag} (K false) |
|
574 *} |
|
575 |
|
576 text {* Now the user can refer to @{attribute my_flag} in |
|
577 declarations, while ML tools can retrieve the current value from the |
|
578 context via @{ML Config.get}. *} |
|
579 |
|
580 ML_val {* @{assert} (Config.get @{context} my_flag = false) *} |
|
581 |
|
582 declare [[my_flag = true]] |
|
583 |
|
584 ML_val {* @{assert} (Config.get @{context} my_flag = true) *} |
|
585 |
|
586 notepad |
|
587 begin |
|
588 { |
|
589 note [[my_flag = false]] |
|
590 ML_val {* @{assert} (Config.get @{context} my_flag = false) *} |
|
591 } |
|
592 ML_val {* @{assert} (Config.get @{context} my_flag = true) *} |
|
593 end |
|
594 |
|
595 text {* Here is another example involving ML type @{ML_type real} |
|
596 (floating-point numbers). *} |
|
597 |
|
598 ML {* |
|
599 val airspeed_velocity = |
|
600 Attrib.setup_config_real @{binding airspeed_velocity} (K 0.0) |
|
601 *} |
|
602 |
|
603 declare [[airspeed_velocity = 10]] |
|
604 declare [[airspeed_velocity = 9.9]] |
|
605 |
|
606 |
|
607 section {* Names \label{sec:names} *} |
|
608 |
|
609 text {* In principle, a name is just a string, but there are various |
|
610 conventions for representing additional structure. For example, |
|
611 ``@{text "Foo.bar.baz"}'' is considered as a long name consisting of |
|
612 qualifier @{text "Foo.bar"} and base name @{text "baz"}. The |
|
613 individual constituents of a name may have further substructure, |
|
614 e.g.\ the string ``\verb,\,\verb,<alpha>,'' encodes as a single |
|
615 symbol (\secref{sec:symbols}). |
|
616 |
|
617 \medskip Subsequently, we shall introduce specific categories of |
|
618 names. Roughly speaking these correspond to logical entities as |
|
619 follows: |
|
620 \begin{itemize} |
|
621 |
|
622 \item Basic names (\secref{sec:basic-name}): free and bound |
|
623 variables. |
|
624 |
|
625 \item Indexed names (\secref{sec:indexname}): schematic variables. |
|
626 |
|
627 \item Long names (\secref{sec:long-name}): constants of any kind |
|
628 (type constructors, term constants, other concepts defined in user |
|
629 space). Such entities are typically managed via name spaces |
|
630 (\secref{sec:name-space}). |
|
631 |
|
632 \end{itemize} |
|
633 *} |
|
634 |
|
635 |
|
636 subsection {* Basic names \label{sec:basic-name} *} |
|
637 |
|
638 text {* |
|
639 A \emph{basic name} essentially consists of a single Isabelle |
|
640 identifier. There are conventions to mark separate classes of basic |
|
641 names, by attaching a suffix of underscores: one underscore means |
|
642 \emph{internal name}, two underscores means \emph{Skolem name}, |
|
643 three underscores means \emph{internal Skolem name}. |
|
644 |
|
645 For example, the basic name @{text "foo"} has the internal version |
|
646 @{text "foo_"}, with Skolem versions @{text "foo__"} and @{text |
|
647 "foo___"}, respectively. |
|
648 |
|
649 These special versions provide copies of the basic name space, apart |
|
650 from anything that normally appears in the user text. For example, |
|
651 system generated variables in Isar proof contexts are usually marked |
|
652 as internal, which prevents mysterious names like @{text "xaa"} to |
|
653 appear in human-readable text. |
|
654 |
|
655 \medskip Manipulating binding scopes often requires on-the-fly |
|
656 renamings. A \emph{name context} contains a collection of already |
|
657 used names. The @{text "declare"} operation adds names to the |
|
658 context. |
|
659 |
|
660 The @{text "invents"} operation derives a number of fresh names from |
|
661 a given starting point. For example, the first three names derived |
|
662 from @{text "a"} are @{text "a"}, @{text "b"}, @{text "c"}. |
|
663 |
|
664 The @{text "variants"} operation produces fresh names by |
|
665 incrementing tentative names as base-26 numbers (with digits @{text |
|
666 "a..z"}) until all clashes are resolved. For example, name @{text |
|
667 "foo"} results in variants @{text "fooa"}, @{text "foob"}, @{text |
|
668 "fooc"}, \dots, @{text "fooaa"}, @{text "fooab"} etc.; each renaming |
|
669 step picks the next unused variant from this sequence. |
|
670 *} |
|
671 |
|
672 text %mlref {* |
|
673 \begin{mldecls} |
|
674 @{index_ML Name.internal: "string -> string"} \\ |
|
675 @{index_ML Name.skolem: "string -> string"} \\ |
|
676 \end{mldecls} |
|
677 \begin{mldecls} |
|
678 @{index_ML_type Name.context} \\ |
|
679 @{index_ML Name.context: Name.context} \\ |
|
680 @{index_ML Name.declare: "string -> Name.context -> Name.context"} \\ |
|
681 @{index_ML Name.invent: "Name.context -> string -> int -> string list"} \\ |
|
682 @{index_ML Name.variant: "string -> Name.context -> string * Name.context"} \\ |
|
683 \end{mldecls} |
|
684 \begin{mldecls} |
|
685 @{index_ML Variable.names_of: "Proof.context -> Name.context"} \\ |
|
686 \end{mldecls} |
|
687 |
|
688 \begin{description} |
|
689 |
|
690 \item @{ML Name.internal}~@{text "name"} produces an internal name |
|
691 by adding one underscore. |
|
692 |
|
693 \item @{ML Name.skolem}~@{text "name"} produces a Skolem name by |
|
694 adding two underscores. |
|
695 |
|
696 \item Type @{ML_type Name.context} represents the context of already |
|
697 used names; the initial value is @{ML "Name.context"}. |
|
698 |
|
699 \item @{ML Name.declare}~@{text "name"} enters a used name into the |
|
700 context. |
|
701 |
|
702 \item @{ML Name.invent}~@{text "context name n"} produces @{text |
|
703 "n"} fresh names derived from @{text "name"}. |
|
704 |
|
705 \item @{ML Name.variant}~@{text "name context"} produces a fresh |
|
706 variant of @{text "name"}; the result is declared to the context. |
|
707 |
|
708 \item @{ML Variable.names_of}~@{text "ctxt"} retrieves the context |
|
709 of declared type and term variable names. Projecting a proof |
|
710 context down to a primitive name context is occasionally useful when |
|
711 invoking lower-level operations. Regular management of ``fresh |
|
712 variables'' is done by suitable operations of structure @{ML_structure |
|
713 Variable}, which is also able to provide an official status of |
|
714 ``locally fixed variable'' within the logical environment (cf.\ |
|
715 \secref{sec:variables}). |
|
716 |
|
717 \end{description} |
|
718 *} |
|
719 |
|
720 text %mlex {* The following simple examples demonstrate how to produce |
|
721 fresh names from the initial @{ML Name.context}. *} |
|
722 |
|
723 ML {* |
|
724 val list1 = Name.invent Name.context "a" 5; |
|
725 @{assert} (list1 = ["a", "b", "c", "d", "e"]); |
|
726 |
|
727 val list2 = |
|
728 #1 (fold_map Name.variant ["x", "x", "a", "a", "'a", "'a"] Name.context); |
|
729 @{assert} (list2 = ["x", "xa", "a", "aa", "'a", "'aa"]); |
|
730 *} |
|
731 |
|
732 text {* \medskip The same works relatively to the formal context as |
|
733 follows. *} |
|
734 |
|
735 locale ex = fixes a b c :: 'a |
|
736 begin |
|
737 |
|
738 ML {* |
|
739 val names = Variable.names_of @{context}; |
|
740 |
|
741 val list1 = Name.invent names "a" 5; |
|
742 @{assert} (list1 = ["d", "e", "f", "g", "h"]); |
|
743 |
|
744 val list2 = |
|
745 #1 (fold_map Name.variant ["x", "x", "a", "a", "'a", "'a"] names); |
|
746 @{assert} (list2 = ["x", "xa", "aa", "ab", "'aa", "'ab"]); |
|
747 *} |
|
748 |
|
749 end |
|
750 |
|
751 |
|
752 subsection {* Indexed names \label{sec:indexname} *} |
|
753 |
|
754 text {* |
|
755 An \emph{indexed name} (or @{text "indexname"}) is a pair of a basic |
|
756 name and a natural number. This representation allows efficient |
|
757 renaming by incrementing the second component only. The canonical |
|
758 way to rename two collections of indexnames apart from each other is |
|
759 this: determine the maximum index @{text "maxidx"} of the first |
|
760 collection, then increment all indexes of the second collection by |
|
761 @{text "maxidx + 1"}; the maximum index of an empty collection is |
|
762 @{text "-1"}. |
|
763 |
|
764 Occasionally, basic names are injected into the same pair type of |
|
765 indexed names: then @{text "(x, -1)"} is used to encode the basic |
|
766 name @{text "x"}. |
|
767 |
|
768 \medskip Isabelle syntax observes the following rules for |
|
769 representing an indexname @{text "(x, i)"} as a packed string: |
|
770 |
|
771 \begin{itemize} |
|
772 |
|
773 \item @{text "?x"} if @{text "x"} does not end with a digit and @{text "i = 0"}, |
|
774 |
|
775 \item @{text "?xi"} if @{text "x"} does not end with a digit, |
|
776 |
|
777 \item @{text "?x.i"} otherwise. |
|
778 |
|
779 \end{itemize} |
|
780 |
|
781 Indexnames may acquire large index numbers after several maxidx |
|
782 shifts have been applied. Results are usually normalized towards |
|
783 @{text "0"} at certain checkpoints, notably at the end of a proof. |
|
784 This works by producing variants of the corresponding basic name |
|
785 components. For example, the collection @{text "?x1, ?x7, ?x42"} |
|
786 becomes @{text "?x, ?xa, ?xb"}. |
|
787 *} |
|
788 |
|
789 text %mlref {* |
|
790 \begin{mldecls} |
|
791 @{index_ML_type indexname: "string * int"} \\ |
|
792 \end{mldecls} |
|
793 |
|
794 \begin{description} |
|
795 |
|
796 \item Type @{ML_type indexname} represents indexed names. This is |
|
797 an abbreviation for @{ML_type "string * int"}. The second component |
|
798 is usually non-negative, except for situations where @{text "(x, |
|
799 -1)"} is used to inject basic names into this type. Other negative |
|
800 indexes should not be used. |
|
801 |
|
802 \end{description} |
|
803 *} |
|
804 |
|
805 |
|
806 subsection {* Long names \label{sec:long-name} *} |
|
807 |
|
808 text {* A \emph{long name} consists of a sequence of non-empty name |
|
809 components. The packed representation uses a dot as separator, as |
|
810 in ``@{text "A.b.c"}''. The last component is called \emph{base |
|
811 name}, the remaining prefix is called \emph{qualifier} (which may be |
|
812 empty). The qualifier can be understood as the access path to the |
|
813 named entity while passing through some nested block-structure, |
|
814 although our free-form long names do not really enforce any strict |
|
815 discipline. |
|
816 |
|
817 For example, an item named ``@{text "A.b.c"}'' may be understood as |
|
818 a local entity @{text "c"}, within a local structure @{text "b"}, |
|
819 within a global structure @{text "A"}. In practice, long names |
|
820 usually represent 1--3 levels of qualification. User ML code should |
|
821 not make any assumptions about the particular structure of long |
|
822 names! |
|
823 |
|
824 The empty name is commonly used as an indication of unnamed |
|
825 entities, or entities that are not entered into the corresponding |
|
826 name space, whenever this makes any sense. The basic operations on |
|
827 long names map empty names again to empty names. |
|
828 *} |
|
829 |
|
830 text %mlref {* |
|
831 \begin{mldecls} |
|
832 @{index_ML Long_Name.base_name: "string -> string"} \\ |
|
833 @{index_ML Long_Name.qualifier: "string -> string"} \\ |
|
834 @{index_ML Long_Name.append: "string -> string -> string"} \\ |
|
835 @{index_ML Long_Name.implode: "string list -> string"} \\ |
|
836 @{index_ML Long_Name.explode: "string -> string list"} \\ |
|
837 \end{mldecls} |
|
838 |
|
839 \begin{description} |
|
840 |
|
841 \item @{ML Long_Name.base_name}~@{text "name"} returns the base name |
|
842 of a long name. |
|
843 |
|
844 \item @{ML Long_Name.qualifier}~@{text "name"} returns the qualifier |
|
845 of a long name. |
|
846 |
|
847 \item @{ML Long_Name.append}~@{text "name\<^sub>1 name\<^sub>2"} appends two long |
|
848 names. |
|
849 |
|
850 \item @{ML Long_Name.implode}~@{text "names"} and @{ML |
|
851 Long_Name.explode}~@{text "name"} convert between the packed string |
|
852 representation and the explicit list form of long names. |
|
853 |
|
854 \end{description} |
|
855 *} |
|
856 |
|
857 |
|
858 subsection {* Name spaces \label{sec:name-space} *} |
|
859 |
|
860 text {* A @{text "name space"} manages a collection of long names, |
|
861 together with a mapping between partially qualified external names |
|
862 and fully qualified internal names (in both directions). Note that |
|
863 the corresponding @{text "intern"} and @{text "extern"} operations |
|
864 are mostly used for parsing and printing only! The @{text |
|
865 "declare"} operation augments a name space according to the accesses |
|
866 determined by a given binding, and a naming policy from the context. |
|
867 |
|
868 \medskip A @{text "binding"} specifies details about the prospective |
|
869 long name of a newly introduced formal entity. It consists of a |
|
870 base name, prefixes for qualification (separate ones for system |
|
871 infrastructure and user-space mechanisms), a slot for the original |
|
872 source position, and some additional flags. |
|
873 |
|
874 \medskip A @{text "naming"} provides some additional details for |
|
875 producing a long name from a binding. Normally, the naming is |
|
876 implicit in the theory or proof context. The @{text "full"} |
|
877 operation (and its variants for different context types) produces a |
|
878 fully qualified internal name to be entered into a name space. The |
|
879 main equation of this ``chemical reaction'' when binding new |
|
880 entities in a context is as follows: |
|
881 |
|
882 \medskip |
|
883 \begin{tabular}{l} |
|
884 @{text "binding + naming \<longrightarrow> long name + name space accesses"} |
|
885 \end{tabular} |
|
886 |
|
887 \bigskip As a general principle, there is a separate name space for |
|
888 each kind of formal entity, e.g.\ fact, logical constant, type |
|
889 constructor, type class. It is usually clear from the occurrence in |
|
890 concrete syntax (or from the scope) which kind of entity a name |
|
891 refers to. For example, the very same name @{text "c"} may be used |
|
892 uniformly for a constant, type constructor, and type class. |
|
893 |
|
894 There are common schemes to name derived entities systematically |
|
895 according to the name of the main logical entity involved, e.g.\ |
|
896 fact @{text "c.intro"} for a canonical introduction rule related to |
|
897 constant @{text "c"}. This technique of mapping names from one |
|
898 space into another requires some care in order to avoid conflicts. |
|
899 In particular, theorem names derived from a type constructor or type |
|
900 class should get an additional suffix in addition to the usual |
|
901 qualification. This leads to the following conventions for derived |
|
902 names: |
|
903 |
|
904 \medskip |
|
905 \begin{tabular}{ll} |
|
906 logical entity & fact name \\\hline |
|
907 constant @{text "c"} & @{text "c.intro"} \\ |
|
908 type @{text "c"} & @{text "c_type.intro"} \\ |
|
909 class @{text "c"} & @{text "c_class.intro"} \\ |
|
910 \end{tabular} |
|
911 *} |
|
912 |
|
913 text %mlref {* |
|
914 \begin{mldecls} |
|
915 @{index_ML_type binding} \\ |
|
916 @{index_ML Binding.empty: binding} \\ |
|
917 @{index_ML Binding.name: "string -> binding"} \\ |
|
918 @{index_ML Binding.qualify: "bool -> string -> binding -> binding"} \\ |
|
919 @{index_ML Binding.prefix: "bool -> string -> binding -> binding"} \\ |
|
920 @{index_ML Binding.conceal: "binding -> binding"} \\ |
|
921 @{index_ML Binding.print: "binding -> string"} \\ |
|
922 \end{mldecls} |
|
923 \begin{mldecls} |
|
924 @{index_ML_type Name_Space.naming} \\ |
|
925 @{index_ML Name_Space.default_naming: Name_Space.naming} \\ |
|
926 @{index_ML Name_Space.add_path: "string -> Name_Space.naming -> Name_Space.naming"} \\ |
|
927 @{index_ML Name_Space.full_name: "Name_Space.naming -> binding -> string"} \\ |
|
928 \end{mldecls} |
|
929 \begin{mldecls} |
|
930 @{index_ML_type Name_Space.T} \\ |
|
931 @{index_ML Name_Space.empty: "string -> Name_Space.T"} \\ |
|
932 @{index_ML Name_Space.merge: "Name_Space.T * Name_Space.T -> Name_Space.T"} \\ |
|
933 @{index_ML Name_Space.declare: "Context.generic -> bool -> |
|
934 binding -> Name_Space.T -> string * Name_Space.T"} \\ |
|
935 @{index_ML Name_Space.intern: "Name_Space.T -> string -> string"} \\ |
|
936 @{index_ML Name_Space.extern: "Proof.context -> Name_Space.T -> string -> string"} \\ |
|
937 @{index_ML Name_Space.is_concealed: "Name_Space.T -> string -> bool"} |
|
938 \end{mldecls} |
|
939 |
|
940 \begin{description} |
|
941 |
|
942 \item Type @{ML_type binding} represents the abstract concept of |
|
943 name bindings. |
|
944 |
|
945 \item @{ML Binding.empty} is the empty binding. |
|
946 |
|
947 \item @{ML Binding.name}~@{text "name"} produces a binding with base |
|
948 name @{text "name"}. Note that this lacks proper source position |
|
949 information; see also the ML antiquotation @{ML_antiquotation |
|
950 binding}. |
|
951 |
|
952 \item @{ML Binding.qualify}~@{text "mandatory name binding"} |
|
953 prefixes qualifier @{text "name"} to @{text "binding"}. The @{text |
|
954 "mandatory"} flag tells if this name component always needs to be |
|
955 given in name space accesses --- this is mostly @{text "false"} in |
|
956 practice. Note that this part of qualification is typically used in |
|
957 derived specification mechanisms. |
|
958 |
|
959 \item @{ML Binding.prefix} is similar to @{ML Binding.qualify}, but |
|
960 affects the system prefix. This part of extra qualification is |
|
961 typically used in the infrastructure for modular specifications, |
|
962 notably ``local theory targets'' (see also \chref{ch:local-theory}). |
|
963 |
|
964 \item @{ML Binding.conceal}~@{text "binding"} indicates that the |
|
965 binding shall refer to an entity that serves foundational purposes |
|
966 only. This flag helps to mark implementation details of |
|
967 specification mechanism etc. Other tools should not depend on the |
|
968 particulars of concealed entities (cf.\ @{ML |
|
969 Name_Space.is_concealed}). |
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970 |
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971 \item @{ML Binding.print}~@{text "binding"} produces a string |
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972 representation for human-readable output, together with some formal |
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973 markup that might get used in GUI front-ends, for example. |
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974 |
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975 \item Type @{ML_type Name_Space.naming} represents the abstract |
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976 concept of a naming policy. |
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977 |
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978 \item @{ML Name_Space.default_naming} is the default naming policy. |
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979 In a theory context, this is usually augmented by a path prefix |
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980 consisting of the theory name. |
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981 |
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982 \item @{ML Name_Space.add_path}~@{text "path naming"} augments the |
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983 naming policy by extending its path component. |
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984 |
|
985 \item @{ML Name_Space.full_name}~@{text "naming binding"} turns a |
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986 name binding (usually a basic name) into the fully qualified |
|
987 internal name, according to the given naming policy. |
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988 |
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989 \item Type @{ML_type Name_Space.T} represents name spaces. |
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990 |
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991 \item @{ML Name_Space.empty}~@{text "kind"} and @{ML Name_Space.merge}~@{text |
|
992 "(space\<^sub>1, space\<^sub>2)"} are the canonical operations for |
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993 maintaining name spaces according to theory data management |
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994 (\secref{sec:context-data}); @{text "kind"} is a formal comment |
|
995 to characterize the purpose of a name space. |
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996 |
|
997 \item @{ML Name_Space.declare}~@{text "context strict binding |
|
998 space"} enters a name binding as fully qualified internal name into |
|
999 the name space, using the naming of the context. |
|
1000 |
|
1001 \item @{ML Name_Space.intern}~@{text "space name"} internalizes a |
|
1002 (partially qualified) external name. |
|
1003 |
|
1004 This operation is mostly for parsing! Note that fully qualified |
|
1005 names stemming from declarations are produced via @{ML |
|
1006 "Name_Space.full_name"} and @{ML "Name_Space.declare"} |
|
1007 (or their derivatives for @{ML_type theory} and |
|
1008 @{ML_type Proof.context}). |
|
1009 |
|
1010 \item @{ML Name_Space.extern}~@{text "ctxt space name"} externalizes a |
|
1011 (fully qualified) internal name. |
|
1012 |
|
1013 This operation is mostly for printing! User code should not rely on |
|
1014 the precise result too much. |
|
1015 |
|
1016 \item @{ML Name_Space.is_concealed}~@{text "space name"} indicates |
|
1017 whether @{text "name"} refers to a strictly private entity that |
|
1018 other tools are supposed to ignore! |
|
1019 |
|
1020 \end{description} |
|
1021 *} |
|
1022 |
|
1023 text %mlantiq {* |
|
1024 \begin{matharray}{rcl} |
|
1025 @{ML_antiquotation_def "binding"} & : & @{text ML_antiquotation} \\ |
|
1026 \end{matharray} |
|
1027 |
|
1028 @{rail \<open> |
|
1029 @@{ML_antiquotation binding} name |
|
1030 \<close>} |
|
1031 |
|
1032 \begin{description} |
|
1033 |
|
1034 \item @{text "@{binding name}"} produces a binding with base name |
|
1035 @{text "name"} and the source position taken from the concrete |
|
1036 syntax of this antiquotation. In many situations this is more |
|
1037 appropriate than the more basic @{ML Binding.name} function. |
|
1038 |
|
1039 \end{description} |
|
1040 *} |
|
1041 |
|
1042 text %mlex {* The following example yields the source position of some |
|
1043 concrete binding inlined into the text: |
|
1044 *} |
|
1045 |
|
1046 ML {* Binding.pos_of @{binding here} *} |
|
1047 |
|
1048 text {* \medskip That position can be also printed in a message as |
|
1049 follows: *} |
|
1050 |
|
1051 ML_command {* |
|
1052 writeln |
|
1053 ("Look here" ^ Position.here (Binding.pos_of @{binding here})) |
|
1054 *} |
|
1055 |
|
1056 text {* This illustrates a key virtue of formalized bindings as |
|
1057 opposed to raw specifications of base names: the system can use this |
|
1058 additional information for feedback given to the user (error |
|
1059 messages etc.). |
|
1060 |
|
1061 \medskip The following example refers to its source position |
|
1062 directly, which is occasionally useful for experimentation and |
|
1063 diagnostic purposes: *} |
|
1064 |
|
1065 ML_command {* |
|
1066 warning ("Look here" ^ Position.here @{here}) |
|
1067 *} |
|
1068 |
|
1069 end |
|