1 (*<*) |
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2 theory Basics |
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3 imports Main |
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4 begin |
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5 (*>*) |
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6 text{* |
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7 This chapter introduces HOL as a functional programming language and shows |
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8 how to prove properties of functional programs by induction. |
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9 |
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10 \section{Basics} |
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11 |
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12 \subsection{Types, Terms and Formulae} |
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13 \label{sec:TypesTermsForms} |
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14 |
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15 HOL is a typed logic whose type system resembles that of functional |
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16 programming languages. Thus there are |
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17 \begin{description} |
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18 \item[base types,] |
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19 in particular @{typ bool}, the type of truth values, |
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20 @{typ nat}, the type of natural numbers ($\mathbb{N}$), and @{typ int}, |
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21 the type of mathematical integers ($\mathbb{Z}$). |
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22 \item[type constructors,] |
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23 in particular @{text list}, the type of |
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24 lists, and @{text set}, the type of sets. Type constructors are written |
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25 postfix, e.g.\ @{typ "nat list"} is the type of lists whose elements are |
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26 natural numbers. |
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27 \item[function types,] |
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28 denoted by @{text"\<Rightarrow>"}. |
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29 \item[type variables,] |
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30 denoted by @{typ 'a}, @{typ 'b} etc., just like in ML\@. |
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31 \end{description} |
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32 |
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33 \concept{Terms} are formed as in functional programming by |
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34 applying functions to arguments. If @{text f} is a function of type |
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35 @{text"\<tau>\<^isub>1 \<Rightarrow> \<tau>\<^isub>2"} and @{text t} is a term of type |
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36 @{text"\<tau>\<^isub>1"} then @{term"f t"} is a term of type @{text"\<tau>\<^isub>2"}. We write @{text "t :: \<tau>"} to mean that term @{text t} has type @{text \<tau>}. |
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37 |
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38 \begin{warn} |
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39 There are many predefined infix symbols like @{text "+"} and @{text"\<le>"}. |
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40 The name of the corresponding binary function is @{term"op +"}, |
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41 not just @{text"+"}. That is, @{term"x + y"} is syntactic sugar for |
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42 \noquotes{@{term[source]"op + x y"}}. |
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43 \end{warn} |
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44 |
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45 HOL also supports some basic constructs from functional programming: |
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46 \begin{quote} |
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47 @{text "(if b then t\<^isub>1 else t\<^isub>2)"}\\ |
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48 @{text "(let x = t in u)"}\\ |
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49 @{text "(case t of pat\<^isub>1 \<Rightarrow> t\<^isub>1 | \<dots> | pat\<^isub>n \<Rightarrow> t\<^isub>n)"} |
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50 \end{quote} |
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51 \begin{warn} |
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52 The above three constructs must always be enclosed in parentheses |
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53 if they occur inside other constructs. |
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54 \end{warn} |
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55 Terms may also contain @{text "\<lambda>"}-abstractions. For example, |
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56 @{term "\<lambda>x. x"} is the identity function. |
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57 |
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58 \concept{Formulae} are terms of type @{text bool}. |
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59 There are the basic constants @{term True} and @{term False} and |
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60 the usual logical connectives (in decreasing order of precedence): |
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61 @{text"\<not>"}, @{text"\<and>"}, @{text"\<or>"}, @{text"\<longrightarrow>"}. |
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62 |
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63 \concept{Equality} is available in the form of the infix function @{text "="} |
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64 of type @{typ "'a \<Rightarrow> 'a \<Rightarrow> bool"}. It also works for formulas, where |
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65 it means ``if and only if''. |
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66 |
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67 \concept{Quantifiers} are written @{prop"\<forall>x. P"} and @{prop"\<exists>x. P"}. |
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68 |
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69 Isabelle automatically computes the type of each variable in a term. This is |
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70 called \concept{type inference}. Despite type inference, it is sometimes |
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71 necessary to attach explicit \concept{type constraints} (or \concept{type |
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72 annotations}) to a variable or term. The syntax is @{text "t :: \<tau>"} as in |
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73 \mbox{\noquotes{@{prop[source] "m < (n::nat)"}}}. Type constraints may be |
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74 needed to |
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75 disambiguate terms involving overloaded functions such as @{text "+"}, @{text |
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76 "*"} and @{text"\<le>"}. |
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77 |
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78 Finally there are the universal quantifier @{text"\<And>"} and the implication |
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79 @{text"\<Longrightarrow>"}. They are part of the Isabelle framework, not the logic |
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80 HOL. Logically, they agree with their HOL counterparts @{text"\<forall>"} and |
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81 @{text"\<longrightarrow>"}, but operationally they behave differently. This will become |
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82 clearer as we go along. |
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83 \begin{warn} |
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84 Right-arrows of all kinds always associate to the right. In particular, |
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85 the formula |
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86 @{text"A\<^isub>1 \<Longrightarrow> A\<^isub>2 \<Longrightarrow> A\<^isub>3"} means @{text "A\<^isub>1 \<Longrightarrow> (A\<^isub>2 \<Longrightarrow> A\<^isub>3)"}. |
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87 The (Isabelle specific) notation \mbox{@{text"\<lbrakk> A\<^isub>1; \<dots>; A\<^isub>n \<rbrakk> \<Longrightarrow> A"}} |
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88 is short for the iterated implication \mbox{@{text"A\<^isub>1 \<Longrightarrow> \<dots> \<Longrightarrow> A\<^isub>n \<Longrightarrow> A"}}. |
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89 Sometimes we also employ inference rule notation: |
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90 \inferrule{\mbox{@{text "A\<^isub>1"}}\\ \mbox{@{text "\<dots>"}}\\ \mbox{@{text "A\<^isub>n"}}} |
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91 {\mbox{@{text A}}} |
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92 \end{warn} |
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93 |
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94 |
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95 \subsection{Theories} |
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96 \label{sec:Basic:Theories} |
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97 |
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98 Roughly speaking, a \concept{theory} is a named collection of types, |
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99 functions, and theorems, much like a module in a programming language. |
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100 All the Isabelle text that you ever type needs to go into a theory. |
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101 The general format of a theory @{text T} is |
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102 \begin{quote} |
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103 \isacom{theory} @{text T}\\ |
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104 \isacom{imports} @{text "T\<^isub>1 \<dots> T\<^isub>n"}\\ |
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105 \isacom{begin}\\ |
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106 \emph{definitions, theorems and proofs}\\ |
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107 \isacom{end} |
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108 \end{quote} |
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109 where @{text "T\<^isub>1 \<dots> T\<^isub>n"} are the names of existing |
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110 theories that @{text T} is based on. The @{text "T\<^isub>i"} are the |
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111 direct \concept{parent theories} of @{text T}. |
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112 Everything defined in the parent theories (and their parents, recursively) is |
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113 automatically visible. Each theory @{text T} must |
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114 reside in a \concept{theory file} named @{text "T.thy"}. |
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115 |
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116 \begin{warn} |
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117 HOL contains a theory @{text Main}, the union of all the basic |
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118 predefined theories like arithmetic, lists, sets, etc. |
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119 Unless you know what you are doing, always include @{text Main} |
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120 as a direct or indirect parent of all your theories. |
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121 \end{warn} |
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122 |
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123 In addition to the theories that come with the Isabelle/HOL distribution |
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124 (see \url{http://isabelle.in.tum.de/library/HOL/}) |
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125 there is also the \emph{Archive of Formal Proofs} |
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126 at \url{http://afp.sourceforge.net}, a growing collection of Isabelle theories |
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127 that everybody can contribute to. |
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128 |
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129 \subsection{Quotation Marks} |
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130 |
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131 The textual definition of a theory follows a fixed syntax with keywords like |
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132 \isacommand{begin} and \isacommand{datatype}. Embedded in this syntax are |
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133 the types and formulae of HOL. To distinguish the two levels, everything |
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134 HOL-specific (terms and types) must be enclosed in quotation marks: |
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135 \texttt{"}\dots\texttt{"}. To lessen this burden, quotation marks around a |
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136 single identifier can be dropped. When Isabelle prints a syntax error |
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137 message, it refers to the HOL syntax as the \concept{inner syntax} and the |
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138 enclosing theory language as the \concept{outer syntax}. |
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139 \sem |
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140 \begin{warn} |
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141 For reasons of readability, we almost never show the quotation marks in this |
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142 book. Consult the accompanying theory files to see where they need to go. |
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143 \end{warn} |
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144 \endsem |
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145 % |
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146 *} |
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147 (*<*) |
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148 end |
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149 (*>*) |
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