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1 theory Framework |
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2 imports Base Main |
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3 begin |
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4 |
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5 chapter {* The Isabelle/Isar Framework \label{ch:isar-framework} *} |
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6 |
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7 text {* |
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8 Isabelle/Isar |
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9 \cite{Wenzel:1999:TPHOL,Wenzel-PhD,Nipkow-TYPES02,Wenzel-Paulson:2006,Wenzel:2006:Festschrift} |
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10 is intended as a generic framework for developing formal |
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11 mathematical documents with full proof checking. Definitions and |
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12 proofs are organized as theories. An assembly of theory sources may |
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13 be presented as a printed document; see also |
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14 \chref{ch:document-prep}. |
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15 |
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16 The main objective of Isar is the design of a human-readable |
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17 structured proof language, which is called the ``primary proof |
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18 format'' in Isar terminology. Such a primary proof language is |
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19 somewhere in the middle between the extremes of primitive proof |
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20 objects and actual natural language. In this respect, Isar is a bit |
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21 more formalistic than Mizar |
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22 \cite{Trybulec:1993:MizarFeatures,Rudnicki:1992:MizarOverview,Wiedijk:1999:Mizar}, |
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23 using logical symbols for certain reasoning schemes where Mizar |
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24 would prefer English words; see \cite{Wenzel-Wiedijk:2002} for |
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25 further comparisons of these systems. |
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26 |
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27 So Isar challenges the traditional way of recording informal proofs |
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28 in mathematical prose, as well as the common tendency to see fully |
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29 formal proofs directly as objects of some logical calculus (e.g.\ |
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30 @{text "\<lambda>"}-terms in a version of type theory). In fact, Isar is |
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31 better understood as an interpreter of a simple block-structured |
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32 language for describing the data flow of local facts and goals, |
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33 interspersed with occasional invocations of proof methods. |
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34 Everything is reduced to logical inferences internally, but these |
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35 steps are somewhat marginal compared to the overall bookkeeping of |
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36 the interpretation process. Thanks to careful design of the syntax |
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37 and semantics of Isar language elements, a formal record of Isar |
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38 instructions may later appear as an intelligible text to the |
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39 attentive reader. |
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40 |
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41 The Isar proof language has emerged from careful analysis of some |
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42 inherent virtues of the existing logical framework of Isabelle/Pure |
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43 \cite{paulson-found,paulson700}, notably composition of higher-order |
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44 natural deduction rules, which is a generalization of Gentzen's |
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45 original calculus \cite{Gentzen:1935}. The approach of generic |
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46 inference systems in Pure is continued by Isar towards actual proof |
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47 texts. |
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48 |
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49 Concrete applications require another intermediate layer: an |
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50 object-logic. Isabelle/HOL \cite{isa-tutorial} (simply-typed |
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51 set-theory) is being used most of the time; Isabelle/ZF |
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52 \cite{isabelle-ZF} is less extensively developed, although it would |
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53 probably fit better for classical mathematics. |
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54 |
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55 \medskip In order to illustrate natural deduction in Isar, we shall |
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56 refer to the background theory and library of Isabelle/HOL. This |
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57 includes common notions of predicate logic, naive set-theory etc.\ |
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58 using fairly standard mathematical notation. From the perspective |
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59 of generic natural deduction there is nothing special about the |
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60 logical connectives of HOL (@{text "\<and>"}, @{text "\<or>"}, @{text "\<forall>"}, |
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61 @{text "\<exists>"}, etc.), only the resulting reasoning principles are |
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62 relevant to the user. There are similar rules available for |
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63 set-theory operators (@{text "\<inter>"}, @{text "\<union>"}, @{text "\<Inter>"}, @{text |
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64 "\<Union>"}, etc.), or any other theory developed in the library (lattice |
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65 theory, topology etc.). |
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66 |
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67 Subsequently we briefly review fragments of Isar proof texts |
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68 corresponding directly to such general deduction schemes. The |
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69 examples shall refer to set-theory, to minimize the danger of |
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70 understanding connectives of predicate logic as something special. |
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71 |
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72 \medskip The following deduction performs @{text "\<inter>"}-introduction, |
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73 working forwards from assumptions towards the conclusion. We give |
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74 both the Isar text, and depict the primitive rule involved, as |
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75 determined by unification of the problem against rules that are |
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76 declared in the library context. |
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77 *} |
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78 |
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79 text_raw {*\medskip\begin{minipage}{0.6\textwidth}*} |
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80 |
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81 (*<*) |
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82 notepad |
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83 begin |
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84 (*>*) |
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85 assume "x \<in> A" and "x \<in> B" |
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86 then have "x \<in> A \<inter> B" .. |
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87 (*<*) |
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88 end |
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89 (*>*) |
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90 |
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91 text_raw {*\end{minipage}\begin{minipage}{0.4\textwidth}*} |
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92 |
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93 text {* |
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94 \infer{@{prop "x \<in> A \<inter> B"}}{@{prop "x \<in> A"} & @{prop "x \<in> B"}} |
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95 *} |
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96 |
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97 text_raw {*\end{minipage}*} |
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98 |
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99 text {* |
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100 \medskip\noindent Note that @{command assume} augments the proof |
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101 context, @{command then} indicates that the current fact shall be |
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102 used in the next step, and @{command have} states an intermediate |
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103 goal. The two dots ``@{command ".."}'' refer to a complete proof of |
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104 this claim, using the indicated facts and a canonical rule from the |
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105 context. We could have been more explicit here by spelling out the |
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106 final proof step via the @{command "by"} command: |
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107 *} |
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108 |
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109 (*<*) |
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110 notepad |
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111 begin |
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112 (*>*) |
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113 assume "x \<in> A" and "x \<in> B" |
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114 then have "x \<in> A \<inter> B" by (rule IntI) |
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115 (*<*) |
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116 end |
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117 (*>*) |
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118 |
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119 text {* |
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120 \noindent The format of the @{text "\<inter>"}-introduction rule represents |
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121 the most basic inference, which proceeds from given premises to a |
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122 conclusion, without any nested proof context involved. |
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123 |
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124 The next example performs backwards introduction on @{term "\<Inter>\<A>"}, |
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125 the intersection of all sets within a given set. This requires a |
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126 nested proof of set membership within a local context, where @{term |
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127 A} is an arbitrary-but-fixed member of the collection: |
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128 *} |
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129 |
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130 text_raw {*\medskip\begin{minipage}{0.6\textwidth}*} |
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131 |
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132 (*<*) |
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133 notepad |
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134 begin |
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135 (*>*) |
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136 have "x \<in> \<Inter>\<A>" |
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137 proof |
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138 fix A |
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139 assume "A \<in> \<A>" |
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140 show "x \<in> A" sorry %noproof |
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141 qed |
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142 (*<*) |
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143 end |
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144 (*>*) |
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145 |
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146 text_raw {*\end{minipage}\begin{minipage}{0.4\textwidth}*} |
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147 |
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148 text {* |
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149 \infer{@{prop "x \<in> \<Inter>\<A>"}}{\infer*{@{prop "x \<in> A"}}{@{text "[A][A \<in> \<A>]"}}} |
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150 *} |
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151 |
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152 text_raw {*\end{minipage}*} |
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153 |
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154 text {* |
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155 \medskip\noindent This Isar reasoning pattern again refers to the |
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156 primitive rule depicted above. The system determines it in the |
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157 ``@{command proof}'' step, which could have been spelt out more |
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158 explicitly as ``@{command proof}~@{text "(rule InterI)"}''. Note |
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159 that the rule involves both a local parameter @{term "A"} and an |
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160 assumption @{prop "A \<in> \<A>"} in the nested reasoning. This kind of |
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161 compound rule typically demands a genuine sub-proof in Isar, working |
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162 backwards rather than forwards as seen before. In the proof body we |
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163 encounter the @{command fix}-@{command assume}-@{command show} |
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164 outline of nested sub-proofs that is typical for Isar. The final |
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165 @{command show} is like @{command have} followed by an additional |
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166 refinement of the enclosing claim, using the rule derived from the |
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167 proof body. |
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168 |
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169 \medskip The next example involves @{term "\<Union>\<A>"}, which can be |
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170 characterized as the set of all @{term "x"} such that @{prop "\<exists>A. x |
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171 \<in> A \<and> A \<in> \<A>"}. The elimination rule for @{prop "x \<in> \<Union>\<A>"} does |
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172 not mention @{text "\<exists>"} and @{text "\<and>"} at all, but admits to obtain |
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173 directly a local @{term "A"} such that @{prop "x \<in> A"} and @{prop "A |
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174 \<in> \<A>"} hold. This corresponds to the following Isar proof and |
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175 inference rule, respectively: |
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176 *} |
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177 |
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178 text_raw {*\medskip\begin{minipage}{0.6\textwidth}*} |
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179 |
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180 (*<*) |
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181 notepad |
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182 begin |
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183 (*>*) |
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184 assume "x \<in> \<Union>\<A>" |
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185 then have C |
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186 proof |
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187 fix A |
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188 assume "x \<in> A" and "A \<in> \<A>" |
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189 show C sorry %noproof |
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190 qed |
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191 (*<*) |
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192 end |
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193 (*>*) |
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194 |
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195 text_raw {*\end{minipage}\begin{minipage}{0.4\textwidth}*} |
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196 |
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197 text {* |
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198 \infer{@{prop "C"}}{@{prop "x \<in> \<Union>\<A>"} & \infer*{@{prop "C"}~}{@{text "[A][x \<in> A, A \<in> \<A>]"}}} |
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199 *} |
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200 |
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201 text_raw {*\end{minipage}*} |
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202 |
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203 text {* |
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204 \medskip\noindent Although the Isar proof follows the natural |
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205 deduction rule closely, the text reads not as natural as |
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206 anticipated. There is a double occurrence of an arbitrary |
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207 conclusion @{prop "C"}, which represents the final result, but is |
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208 irrelevant for now. This issue arises for any elimination rule |
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209 involving local parameters. Isar provides the derived language |
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210 element @{command obtain}, which is able to perform the same |
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211 elimination proof more conveniently: |
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212 *} |
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213 |
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214 (*<*) |
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215 notepad |
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216 begin |
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217 (*>*) |
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218 assume "x \<in> \<Union>\<A>" |
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219 then obtain A where "x \<in> A" and "A \<in> \<A>" .. |
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220 (*<*) |
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221 end |
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222 (*>*) |
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223 |
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224 text {* |
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225 \noindent Here we avoid to mention the final conclusion @{prop "C"} |
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226 and return to plain forward reasoning. The rule involved in the |
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227 ``@{command ".."}'' proof is the same as before. |
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228 *} |
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229 |
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230 |
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231 section {* The Pure framework \label{sec:framework-pure} *} |
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232 |
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233 text {* |
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234 The Pure logic \cite{paulson-found,paulson700} is an intuitionistic |
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235 fragment of higher-order logic \cite{church40}. In type-theoretic |
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236 parlance, there are three levels of @{text "\<lambda>"}-calculus with |
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237 corresponding arrows @{text "\<Rightarrow>"}/@{text "\<And>"}/@{text "\<Longrightarrow>"}: |
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238 |
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239 \medskip |
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240 \begin{tabular}{ll} |
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241 @{text "\<alpha> \<Rightarrow> \<beta>"} & syntactic function space (terms depending on terms) \\ |
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242 @{text "\<And>x. B(x)"} & universal quantification (proofs depending on terms) \\ |
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243 @{text "A \<Longrightarrow> B"} & implication (proofs depending on proofs) \\ |
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244 \end{tabular} |
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245 \medskip |
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246 |
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247 \noindent Here only the types of syntactic terms, and the |
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248 propositions of proof terms have been shown. The @{text |
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249 "\<lambda>"}-structure of proofs can be recorded as an optional feature of |
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250 the Pure inference kernel \cite{Berghofer-Nipkow:2000:TPHOL}, but |
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251 the formal system can never depend on them due to \emph{proof |
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252 irrelevance}. |
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253 |
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254 On top of this most primitive layer of proofs, Pure implements a |
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255 generic calculus for nested natural deduction rules, similar to |
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256 \cite{Schroeder-Heister:1984}. Here object-logic inferences are |
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257 internalized as formulae over @{text "\<And>"} and @{text "\<Longrightarrow>"}. |
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258 Combining such rule statements may involve higher-order unification |
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259 \cite{paulson-natural}. |
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260 *} |
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261 |
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262 |
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263 subsection {* Primitive inferences *} |
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264 |
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265 text {* |
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266 Term syntax provides explicit notation for abstraction @{text "\<lambda>x :: |
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267 \<alpha>. b(x)"} and application @{text "b a"}, while types are usually |
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268 implicit thanks to type-inference; terms of type @{text "prop"} are |
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269 called propositions. Logical statements are composed via @{text "\<And>x |
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270 :: \<alpha>. B(x)"} and @{text "A \<Longrightarrow> B"}. Primitive reasoning operates on |
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271 judgments of the form @{text "\<Gamma> \<turnstile> \<phi>"}, with standard introduction |
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272 and elimination rules for @{text "\<And>"} and @{text "\<Longrightarrow>"} that refer to |
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273 fixed parameters @{text "x\<^isub>1, \<dots>, x\<^isub>m"} and hypotheses |
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274 @{text "A\<^isub>1, \<dots>, A\<^isub>n"} from the context @{text "\<Gamma>"}; |
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275 the corresponding proof terms are left implicit. The subsequent |
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276 inference rules define @{text "\<Gamma> \<turnstile> \<phi>"} inductively, relative to a |
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277 collection of axioms: |
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278 |
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279 \[ |
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280 \infer{@{text "\<turnstile> A"}}{(@{text "A"} \text{~axiom})} |
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281 \qquad |
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282 \infer{@{text "A \<turnstile> A"}}{} |
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283 \] |
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284 |
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285 \[ |
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286 \infer{@{text "\<Gamma> \<turnstile> \<And>x. B(x)"}}{@{text "\<Gamma> \<turnstile> B(x)"} & @{text "x \<notin> \<Gamma>"}} |
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287 \qquad |
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288 \infer{@{text "\<Gamma> \<turnstile> B(a)"}}{@{text "\<Gamma> \<turnstile> \<And>x. B(x)"}} |
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289 \] |
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290 |
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291 \[ |
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292 \infer{@{text "\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}} |
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293 \qquad |
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294 \infer{@{text "\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B"}}{@{text "\<Gamma>\<^sub>1 \<turnstile> A \<Longrightarrow> B"} & @{text "\<Gamma>\<^sub>2 \<turnstile> A"}} |
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295 \] |
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296 |
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297 Furthermore, Pure provides a built-in equality @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> |
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298 prop"} with axioms for reflexivity, substitution, extensionality, |
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299 and @{text "\<alpha>\<beta>\<eta>"}-conversion on @{text "\<lambda>"}-terms. |
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300 |
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301 \medskip An object-logic introduces another layer on top of Pure, |
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302 e.g.\ with types @{text "i"} for individuals and @{text "o"} for |
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303 propositions, term constants @{text "Trueprop :: o \<Rightarrow> prop"} as |
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304 (implicit) derivability judgment and connectives like @{text "\<and> :: o |
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305 \<Rightarrow> o \<Rightarrow> o"} or @{text "\<forall> :: (i \<Rightarrow> o) \<Rightarrow> o"}, and axioms for object-level |
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306 rules such as @{text "conjI: A \<Longrightarrow> B \<Longrightarrow> A \<and> B"} or @{text "allI: (\<And>x. B |
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307 x) \<Longrightarrow> \<forall>x. B x"}. Derived object rules are represented as theorems of |
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308 Pure. After the initial object-logic setup, further axiomatizations |
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309 are usually avoided; plain definitions and derived principles are |
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310 used exclusively. |
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311 *} |
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312 |
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313 |
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314 subsection {* Reasoning with rules \label{sec:framework-resolution} *} |
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315 |
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316 text {* |
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317 Primitive inferences mostly serve foundational purposes. The main |
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318 reasoning mechanisms of Pure operate on nested natural deduction |
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319 rules expressed as formulae, using @{text "\<And>"} to bind local |
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320 parameters and @{text "\<Longrightarrow>"} to express entailment. Multiple |
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321 parameters and premises are represented by repeating these |
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322 connectives in a right-associative manner. |
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323 |
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324 Since @{text "\<And>"} and @{text "\<Longrightarrow>"} commute thanks to the theorem |
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325 @{prop "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}, we may assume w.l.o.g.\ |
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326 that rule statements always observe the normal form where |
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327 quantifiers are pulled in front of implications at each level of |
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328 nesting. This means that any Pure proposition may be presented as a |
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329 \emph{Hereditary Harrop Formula} \cite{Miller:1991} which is of the |
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330 form @{text "\<And>x\<^isub>1 \<dots> x\<^isub>m. H\<^isub>1 \<Longrightarrow> \<dots> H\<^isub>n \<Longrightarrow> |
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331 A"} for @{text "m, n \<ge> 0"}, and @{text "A"} atomic, and @{text |
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332 "H\<^isub>1, \<dots>, H\<^isub>n"} being recursively of the same format. |
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333 Following the convention that outermost quantifiers are implicit, |
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334 Horn clauses @{text "A\<^isub>1 \<Longrightarrow> \<dots> A\<^isub>n \<Longrightarrow> A"} are a special |
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335 case of this. |
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336 |
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337 For example, @{text "\<inter>"}-introduction rule encountered before is |
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338 represented as a Pure theorem as follows: |
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339 \[ |
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340 @{text "IntI:"}~@{prop "x \<in> A \<Longrightarrow> x \<in> B \<Longrightarrow> x \<in> A \<inter> B"} |
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341 \] |
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342 |
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343 \noindent This is a plain Horn clause, since no further nesting on |
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344 the left is involved. The general @{text "\<Inter>"}-introduction |
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345 corresponds to a Hereditary Harrop Formula with one additional level |
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346 of nesting: |
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347 \[ |
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348 @{text "InterI:"}~@{prop "(\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A) \<Longrightarrow> x \<in> \<Inter>\<A>"} |
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349 \] |
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350 |
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351 \medskip Goals are also represented as rules: @{text "A\<^isub>1 \<Longrightarrow> |
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352 \<dots> A\<^isub>n \<Longrightarrow> C"} states that the sub-goals @{text "A\<^isub>1, \<dots>, |
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353 A\<^isub>n"} entail the result @{text "C"}; for @{text "n = 0"} the |
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354 goal is finished. To allow @{text "C"} being a rule statement |
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355 itself, we introduce the protective marker @{text "# :: prop \<Rightarrow> |
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356 prop"}, which is defined as identity and hidden from the user. We |
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357 initialize and finish goal states as follows: |
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358 |
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359 \[ |
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360 \begin{array}{c@ {\qquad}c} |
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361 \infer[(@{inference_def init})]{@{text "C \<Longrightarrow> #C"}}{} & |
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362 \infer[(@{inference_def finish})]{@{text C}}{@{text "#C"}} |
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363 \end{array} |
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364 \] |
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365 |
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366 \noindent Goal states are refined in intermediate proof steps until |
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367 a finished form is achieved. Here the two main reasoning principles |
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368 are @{inference resolution}, for back-chaining a rule against a |
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369 sub-goal (replacing it by zero or more sub-goals), and @{inference |
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370 assumption}, for solving a sub-goal (finding a short-circuit with |
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371 local assumptions). Below @{text "\<^vec>x"} stands for @{text |
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372 "x\<^isub>1, \<dots>, x\<^isub>n"} (@{text "n \<ge> 0"}). |
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373 |
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374 \[ |
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375 \infer[(@{inference_def resolution})] |
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376 {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>A (\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>"}} |
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377 {\begin{tabular}{rl} |
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378 @{text "rule:"} & |
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379 @{text "\<^vec>A \<^vec>a \<Longrightarrow> B \<^vec>a"} \\ |
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380 @{text "goal:"} & |
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381 @{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C"} \\ |
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382 @{text "goal unifier:"} & |
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383 @{text "(\<lambda>\<^vec>x. B (\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>"} \\ |
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384 \end{tabular}} |
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385 \] |
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386 |
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387 \medskip |
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388 |
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389 \[ |
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390 \infer[(@{inference_def assumption})]{@{text "C\<vartheta>"}} |
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391 {\begin{tabular}{rl} |
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392 @{text "goal:"} & |
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393 @{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x) \<Longrightarrow> C"} \\ |
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394 @{text "assm unifier:"} & @{text "A\<vartheta> = H\<^sub>i\<vartheta>"}~~\text{(for some~@{text "H\<^sub>i"})} \\ |
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395 \end{tabular}} |
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396 \] |
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397 |
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398 The following trace illustrates goal-oriented reasoning in |
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399 Isabelle/Pure: |
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400 |
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401 {\footnotesize |
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402 \medskip |
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403 \begin{tabular}{r@ {\quad}l} |
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404 @{text "(A \<and> B \<Longrightarrow> B \<and> A) \<Longrightarrow> #(A \<and> B \<Longrightarrow> B \<and> A)"} & @{text "(init)"} \\ |
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405 @{text "(A \<and> B \<Longrightarrow> B) \<Longrightarrow> (A \<and> B \<Longrightarrow> A) \<Longrightarrow> #\<dots>"} & @{text "(resolution B \<Longrightarrow> A \<Longrightarrow> B \<and> A)"} \\ |
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406 @{text "(A \<and> B \<Longrightarrow> A \<and> B) \<Longrightarrow> (A \<and> B \<Longrightarrow> A) \<Longrightarrow> #\<dots>"} & @{text "(resolution A \<and> B \<Longrightarrow> B)"} \\ |
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407 @{text "(A \<and> B \<Longrightarrow> A) \<Longrightarrow> #\<dots>"} & @{text "(assumption)"} \\ |
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408 @{text "(A \<and> B \<Longrightarrow> A \<and> B) \<Longrightarrow> #\<dots>"} & @{text "(resolution A \<and> B \<Longrightarrow> A)"} \\ |
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409 @{text "#\<dots>"} & @{text "(assumption)"} \\ |
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410 @{text "A \<and> B \<Longrightarrow> B \<and> A"} & @{text "(finish)"} \\ |
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411 \end{tabular} |
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412 \medskip |
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413 } |
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414 |
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415 Compositions of @{inference assumption} after @{inference |
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416 resolution} occurs quite often, typically in elimination steps. |
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417 Traditional Isabelle tactics accommodate this by a combined |
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418 @{inference_def elim_resolution} principle. In contrast, Isar uses |
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419 a slightly more refined combination, where the assumptions to be |
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420 closed are marked explicitly, using again the protective marker |
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421 @{text "#"}: |
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422 |
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423 \[ |
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424 \infer[(@{inference refinement})] |
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425 {@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>G' (\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>"}} |
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426 {\begin{tabular}{rl} |
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427 @{text "sub\<hyphen>proof:"} & |
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428 @{text "\<^vec>G \<^vec>a \<Longrightarrow> B \<^vec>a"} \\ |
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429 @{text "goal:"} & |
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430 @{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C"} \\ |
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431 @{text "goal unifier:"} & |
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432 @{text "(\<lambda>\<^vec>x. B (\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>"} \\ |
|
433 @{text "assm unifiers:"} & |
|
434 @{text "(\<lambda>\<^vec>x. G\<^sub>j (\<^vec>a \<^vec>x))\<vartheta> = #H\<^sub>i\<vartheta>"} \\ |
|
435 & \quad (for each marked @{text "G\<^sub>j"} some @{text "#H\<^sub>i"}) \\ |
|
436 \end{tabular}} |
|
437 \] |
|
438 |
|
439 \noindent Here the @{text "sub\<hyphen>proof"} rule stems from the |
|
440 main @{command fix}-@{command assume}-@{command show} outline of |
|
441 Isar (cf.\ \secref{sec:framework-subproof}): each assumption |
|
442 indicated in the text results in a marked premise @{text "G"} above. |
|
443 The marking enforces resolution against one of the sub-goal's |
|
444 premises. Consequently, @{command fix}-@{command assume}-@{command |
|
445 show} enables to fit the result of a sub-proof quite robustly into a |
|
446 pending sub-goal, while maintaining a good measure of flexibility. |
|
447 *} |
|
448 |
|
449 |
|
450 section {* The Isar proof language \label{sec:framework-isar} *} |
|
451 |
|
452 text {* |
|
453 Structured proofs are presented as high-level expressions for |
|
454 composing entities of Pure (propositions, facts, and goals). The |
|
455 Isar proof language allows to organize reasoning within the |
|
456 underlying rule calculus of Pure, but Isar is not another logical |
|
457 calculus! |
|
458 |
|
459 Isar is an exercise in sound minimalism. Approximately half of the |
|
460 language is introduced as primitive, the rest defined as derived |
|
461 concepts. The following grammar describes the core language |
|
462 (category @{text "proof"}), which is embedded into theory |
|
463 specification elements such as @{command theorem}; see also |
|
464 \secref{sec:framework-stmt} for the separate category @{text |
|
465 "statement"}. |
|
466 |
|
467 \medskip |
|
468 \begin{tabular}{rcl} |
|
469 @{text "theory\<hyphen>stmt"} & = & @{command "theorem"}~@{text "statement proof |"}~~@{command "definition"}~@{text "\<dots> | \<dots>"} \\[1ex] |
|
470 |
|
471 @{text "proof"} & = & @{text "prfx\<^sup>*"}~@{command "proof"}~@{text "method\<^sup>? stmt\<^sup>*"}~@{command "qed"}~@{text "method\<^sup>?"} \\[1ex] |
|
472 |
|
473 @{text prfx} & = & @{command "using"}~@{text "facts"} \\ |
|
474 & @{text "|"} & @{command "unfolding"}~@{text "facts"} \\ |
|
475 |
|
476 @{text stmt} & = & @{command "{"}~@{text "stmt\<^sup>*"}~@{command "}"} \\ |
|
477 & @{text "|"} & @{command "next"} \\ |
|
478 & @{text "|"} & @{command "note"}~@{text "name = facts"} \\ |
|
479 & @{text "|"} & @{command "let"}~@{text "term = term"} \\ |
|
480 & @{text "|"} & @{command "fix"}~@{text "var\<^sup>+"} \\ |
|
481 & @{text "|"} & @{command assume}~@{text "\<guillemotleft>inference\<guillemotright> name: props"} \\ |
|
482 & @{text "|"} & @{command "then"}@{text "\<^sup>?"}~@{text goal} \\ |
|
483 @{text goal} & = & @{command "have"}~@{text "name: props proof"} \\ |
|
484 & @{text "|"} & @{command "show"}~@{text "name: props proof"} \\ |
|
485 \end{tabular} |
|
486 |
|
487 \medskip Simultaneous propositions or facts may be separated by the |
|
488 @{keyword "and"} keyword. |
|
489 |
|
490 \medskip The syntax for terms and propositions is inherited from |
|
491 Pure (and the object-logic). A @{text "pattern"} is a @{text |
|
492 "term"} with schematic variables, to be bound by higher-order |
|
493 matching. |
|
494 |
|
495 \medskip Facts may be referenced by name or proposition. For |
|
496 example, the result of ``@{command have}~@{text "a: A \<langle>proof\<rangle>"}'' |
|
497 becomes available both as @{text "a"} and |
|
498 \isacharbackquoteopen@{text "A"}\isacharbackquoteclose. Moreover, |
|
499 fact expressions may involve attributes that modify either the |
|
500 theorem or the background context. For example, the expression |
|
501 ``@{text "a [OF b]"}'' refers to the composition of two facts |
|
502 according to the @{inference resolution} inference of |
|
503 \secref{sec:framework-resolution}, while ``@{text "a [intro]"}'' |
|
504 declares a fact as introduction rule in the context. |
|
505 |
|
506 The special fact called ``@{fact this}'' always refers to the last |
|
507 result, as produced by @{command note}, @{command assume}, @{command |
|
508 have}, or @{command show}. Since @{command note} occurs |
|
509 frequently together with @{command then} we provide some |
|
510 abbreviations: |
|
511 |
|
512 \medskip |
|
513 \begin{tabular}{rcl} |
|
514 @{command from}~@{text a} & @{text "\<equiv>"} & @{command note}~@{text a}~@{command then} \\ |
|
515 @{command with}~@{text a} & @{text "\<equiv>"} & @{command from}~@{text "a \<AND> this"} \\ |
|
516 \end{tabular} |
|
517 \medskip |
|
518 |
|
519 The @{text "method"} category is essentially a parameter and may be |
|
520 populated later. Methods use the facts indicated by @{command |
|
521 "then"} or @{command using}, and then operate on the goal state. |
|
522 Some basic methods are predefined: ``@{method "-"}'' leaves the goal |
|
523 unchanged, ``@{method this}'' applies the facts as rules to the |
|
524 goal, ``@{method (Pure) "rule"}'' applies the facts to another rule and the |
|
525 result to the goal (both ``@{method this}'' and ``@{method (Pure) rule}'' |
|
526 refer to @{inference resolution} of |
|
527 \secref{sec:framework-resolution}). The secondary arguments to |
|
528 ``@{method (Pure) rule}'' may be specified explicitly as in ``@{text "(rule |
|
529 a)"}'', or picked from the context. In the latter case, the system |
|
530 first tries rules declared as @{attribute (Pure) elim} or |
|
531 @{attribute (Pure) dest}, followed by those declared as @{attribute |
|
532 (Pure) intro}. |
|
533 |
|
534 The default method for @{command proof} is ``@{method (Pure) rule}'' |
|
535 (arguments picked from the context), for @{command qed} it is |
|
536 ``@{method "-"}''. Further abbreviations for terminal proof steps |
|
537 are ``@{command "by"}~@{text "method\<^sub>1 method\<^sub>2"}'' for |
|
538 ``@{command proof}~@{text "method\<^sub>1"}~@{command qed}~@{text |
|
539 "method\<^sub>2"}'', and ``@{command ".."}'' for ``@{command |
|
540 "by"}~@{method (Pure) rule}, and ``@{command "."}'' for ``@{command |
|
541 "by"}~@{method this}''. The @{command unfolding} element operates |
|
542 directly on the current facts and goal by applying equalities. |
|
543 |
|
544 \medskip Block structure can be indicated explicitly by ``@{command |
|
545 "{"}~@{text "\<dots>"}~@{command "}"}'', although the body of a sub-proof |
|
546 already involves implicit nesting. In any case, @{command next} |
|
547 jumps into the next section of a block, i.e.\ it acts like closing |
|
548 an implicit block scope and opening another one; there is no direct |
|
549 correspondence to subgoals here. |
|
550 |
|
551 The remaining elements @{command fix} and @{command assume} build up |
|
552 a local context (see \secref{sec:framework-context}), while |
|
553 @{command show} refines a pending sub-goal by the rule resulting |
|
554 from a nested sub-proof (see \secref{sec:framework-subproof}). |
|
555 Further derived concepts will support calculational reasoning (see |
|
556 \secref{sec:framework-calc}). |
|
557 *} |
|
558 |
|
559 |
|
560 subsection {* Context elements \label{sec:framework-context} *} |
|
561 |
|
562 text {* |
|
563 In judgments @{text "\<Gamma> \<turnstile> \<phi>"} of the primitive framework, @{text "\<Gamma>"} |
|
564 essentially acts like a proof context. Isar elaborates this idea |
|
565 towards a higher-level notion, with additional information for |
|
566 type-inference, term abbreviations, local facts, hypotheses etc. |
|
567 |
|
568 The element @{command fix}~@{text "x :: \<alpha>"} declares a local |
|
569 parameter, i.e.\ an arbitrary-but-fixed entity of a given type; in |
|
570 results exported from the context, @{text "x"} may become anything. |
|
571 The @{command assume}~@{text "\<guillemotleft>inference\<guillemotright>"} element provides a |
|
572 general interface to hypotheses: ``@{command assume}~@{text |
|
573 "\<guillemotleft>inference\<guillemotright> A"}'' produces @{text "A \<turnstile> A"} locally, while the |
|
574 included inference tells how to discharge @{text A} from results |
|
575 @{text "A \<turnstile> B"} later on. There is no user-syntax for @{text |
|
576 "\<guillemotleft>inference\<guillemotright>"}, i.e.\ it may only occur internally when derived |
|
577 commands are defined in ML. |
|
578 |
|
579 At the user-level, the default inference for @{command assume} is |
|
580 @{inference discharge} as given below. The additional variants |
|
581 @{command presume} and @{command def} are defined as follows: |
|
582 |
|
583 \medskip |
|
584 \begin{tabular}{rcl} |
|
585 @{command presume}~@{text A} & @{text "\<equiv>"} & @{command assume}~@{text "\<guillemotleft>weak\<hyphen>discharge\<guillemotright> A"} \\ |
|
586 @{command def}~@{text "x \<equiv> a"} & @{text "\<equiv>"} & @{command fix}~@{text x}~@{command assume}~@{text "\<guillemotleft>expansion\<guillemotright> x \<equiv> a"} \\ |
|
587 \end{tabular} |
|
588 \medskip |
|
589 |
|
590 \[ |
|
591 \infer[(@{inference_def discharge})]{@{text "\<strut>\<Gamma> - A \<turnstile> #A \<Longrightarrow> B"}}{@{text "\<strut>\<Gamma> \<turnstile> B"}} |
|
592 \] |
|
593 \[ |
|
594 \infer[(@{inference_def "weak\<hyphen>discharge"})]{@{text "\<strut>\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<strut>\<Gamma> \<turnstile> B"}} |
|
595 \] |
|
596 \[ |
|
597 \infer[(@{inference_def expansion})]{@{text "\<strut>\<Gamma> - (x \<equiv> a) \<turnstile> B a"}}{@{text "\<strut>\<Gamma> \<turnstile> B x"}} |
|
598 \] |
|
599 |
|
600 \medskip Note that @{inference discharge} and @{inference |
|
601 "weak\<hyphen>discharge"} differ in the marker for @{prop A}, which is |
|
602 relevant when the result of a @{command fix}-@{command |
|
603 assume}-@{command show} outline is composed with a pending goal, |
|
604 cf.\ \secref{sec:framework-subproof}. |
|
605 |
|
606 The most interesting derived context element in Isar is @{command |
|
607 obtain} \cite[\S5.3]{Wenzel-PhD}, which supports generalized |
|
608 elimination steps in a purely forward manner. The @{command obtain} |
|
609 command takes a specification of parameters @{text "\<^vec>x"} and |
|
610 assumptions @{text "\<^vec>A"} to be added to the context, together |
|
611 with a proof of a case rule stating that this extension is |
|
612 conservative (i.e.\ may be removed from closed results later on): |
|
613 |
|
614 \medskip |
|
615 \begin{tabular}{l} |
|
616 @{text "\<langle>facts\<rangle>"}~~@{command obtain}~@{text "\<^vec>x \<WHERE> \<^vec>A \<^vec>x \<langle>proof\<rangle> \<equiv>"} \\[0.5ex] |
|
617 \quad @{command have}~@{text "case: \<And>thesis. (\<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis) \<Longrightarrow> thesis\<rangle>"} \\ |
|
618 \quad @{command proof}~@{method "-"} \\ |
|
619 \qquad @{command fix}~@{text thesis} \\ |
|
620 \qquad @{command assume}~@{text "[intro]: \<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis"} \\ |
|
621 \qquad @{command show}~@{text thesis}~@{command using}~@{text "\<langle>facts\<rangle> \<langle>proof\<rangle>"} \\ |
|
622 \quad @{command qed} \\ |
|
623 \quad @{command fix}~@{text "\<^vec>x"}~@{command assume}~@{text "\<guillemotleft>elimination case\<guillemotright> \<^vec>A \<^vec>x"} \\ |
|
624 \end{tabular} |
|
625 \medskip |
|
626 |
|
627 \[ |
|
628 \infer[(@{inference elimination})]{@{text "\<Gamma> \<turnstile> B"}}{ |
|
629 \begin{tabular}{rl} |
|
630 @{text "case:"} & |
|
631 @{text "\<Gamma> \<turnstile> \<And>thesis. (\<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis) \<Longrightarrow> thesis"} \\[0.2ex] |
|
632 @{text "result:"} & |
|
633 @{text "\<Gamma> \<union> \<^vec>A \<^vec>y \<turnstile> B"} \\[0.2ex] |
|
634 \end{tabular}} |
|
635 \] |
|
636 |
|
637 \noindent Here the name ``@{text thesis}'' is a specific convention |
|
638 for an arbitrary-but-fixed proposition; in the primitive natural |
|
639 deduction rules shown before we have occasionally used @{text C}. |
|
640 The whole statement of ``@{command obtain}~@{text x}~@{keyword |
|
641 "where"}~@{text "A x"}'' may be read as a claim that @{text "A x"} |
|
642 may be assumed for some arbitrary-but-fixed @{text "x"}. Also note |
|
643 that ``@{command obtain}~@{text "A \<AND> B"}'' without parameters |
|
644 is similar to ``@{command have}~@{text "A \<AND> B"}'', but the |
|
645 latter involves multiple sub-goals. |
|
646 |
|
647 \medskip The subsequent Isar proof texts explain all context |
|
648 elements introduced above using the formal proof language itself. |
|
649 After finishing a local proof within a block, we indicate the |
|
650 exported result via @{command note}. |
|
651 *} |
|
652 |
|
653 (*<*) |
|
654 theorem True |
|
655 proof |
|
656 (*>*) |
|
657 txt_raw {* \begin{minipage}[t]{0.45\textwidth} *} |
|
658 { |
|
659 fix x |
|
660 have "B x" sorry %noproof |
|
661 } |
|
662 note `\<And>x. B x` |
|
663 txt_raw {* \end{minipage}\quad\begin{minipage}[t]{0.45\textwidth} *}(*<*)next(*>*) |
|
664 { |
|
665 assume A |
|
666 have B sorry %noproof |
|
667 } |
|
668 note `A \<Longrightarrow> B` |
|
669 txt_raw {* \end{minipage}\\[3ex]\begin{minipage}[t]{0.45\textwidth} *}(*<*)next(*>*) |
|
670 { |
|
671 def x \<equiv> a |
|
672 have "B x" sorry %noproof |
|
673 } |
|
674 note `B a` |
|
675 txt_raw {* \end{minipage}\quad\begin{minipage}[t]{0.45\textwidth} *}(*<*)next(*>*) |
|
676 { |
|
677 obtain x where "A x" sorry %noproof |
|
678 have B sorry %noproof |
|
679 } |
|
680 note `B` |
|
681 txt_raw {* \end{minipage} *} |
|
682 (*<*) |
|
683 qed |
|
684 (*>*) |
|
685 |
|
686 text {* |
|
687 \bigskip\noindent This illustrates the meaning of Isar context |
|
688 elements without goals getting in between. |
|
689 *} |
|
690 |
|
691 subsection {* Structured statements \label{sec:framework-stmt} *} |
|
692 |
|
693 text {* |
|
694 The category @{text "statement"} of top-level theorem specifications |
|
695 is defined as follows: |
|
696 |
|
697 \medskip |
|
698 \begin{tabular}{rcl} |
|
699 @{text "statement"} & @{text "\<equiv>"} & @{text "name: props \<AND> \<dots>"} \\ |
|
700 & @{text "|"} & @{text "context\<^sup>* conclusion"} \\[0.5ex] |
|
701 |
|
702 @{text "context"} & @{text "\<equiv>"} & @{text "\<FIXES> vars \<AND> \<dots>"} \\ |
|
703 & @{text "|"} & @{text "\<ASSUMES> name: props \<AND> \<dots>"} \\ |
|
704 |
|
705 @{text "conclusion"} & @{text "\<equiv>"} & @{text "\<SHOWS> name: props \<AND> \<dots>"} \\ |
|
706 & @{text "|"} & @{text "\<OBTAINS> vars \<AND> \<dots> \<WHERE> name: props \<AND> \<dots>"} \\ |
|
707 & & \quad @{text "\<BBAR> \<dots>"} \\ |
|
708 \end{tabular} |
|
709 |
|
710 \medskip\noindent A simple @{text "statement"} consists of named |
|
711 propositions. The full form admits local context elements followed |
|
712 by the actual conclusions, such as ``@{keyword "fixes"}~@{text |
|
713 x}~@{keyword "assumes"}~@{text "A x"}~@{keyword "shows"}~@{text "B |
|
714 x"}''. The final result emerges as a Pure rule after discharging |
|
715 the context: @{prop "\<And>x. A x \<Longrightarrow> B x"}. |
|
716 |
|
717 The @{keyword "obtains"} variant is another abbreviation defined |
|
718 below; unlike @{command obtain} (cf.\ |
|
719 \secref{sec:framework-context}) there may be several ``cases'' |
|
720 separated by ``@{text "\<BBAR>"}'', each consisting of several |
|
721 parameters (@{text "vars"}) and several premises (@{text "props"}). |
|
722 This specifies multi-branch elimination rules. |
|
723 |
|
724 \medskip |
|
725 \begin{tabular}{l} |
|
726 @{text "\<OBTAINS> \<^vec>x \<WHERE> \<^vec>A \<^vec>x \<BBAR> \<dots> \<equiv>"} \\[0.5ex] |
|
727 \quad @{text "\<FIXES> thesis"} \\ |
|
728 \quad @{text "\<ASSUMES> [intro]: \<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis \<AND> \<dots>"} \\ |
|
729 \quad @{text "\<SHOWS> thesis"} \\ |
|
730 \end{tabular} |
|
731 \medskip |
|
732 |
|
733 Presenting structured statements in such an ``open'' format usually |
|
734 simplifies the subsequent proof, because the outer structure of the |
|
735 problem is already laid out directly. E.g.\ consider the following |
|
736 canonical patterns for @{text "\<SHOWS>"} and @{text "\<OBTAINS>"}, |
|
737 respectively: |
|
738 *} |
|
739 |
|
740 text_raw {*\begin{minipage}{0.5\textwidth}*} |
|
741 |
|
742 theorem |
|
743 fixes x and y |
|
744 assumes "A x" and "B y" |
|
745 shows "C x y" |
|
746 proof - |
|
747 from `A x` and `B y` |
|
748 show "C x y" sorry %noproof |
|
749 qed |
|
750 |
|
751 text_raw {*\end{minipage}\begin{minipage}{0.5\textwidth}*} |
|
752 |
|
753 theorem |
|
754 obtains x and y |
|
755 where "A x" and "B y" |
|
756 proof - |
|
757 have "A a" and "B b" sorry %noproof |
|
758 then show thesis .. |
|
759 qed |
|
760 |
|
761 text_raw {*\end{minipage}*} |
|
762 |
|
763 text {* |
|
764 \medskip\noindent Here local facts \isacharbackquoteopen@{text "A |
|
765 x"}\isacharbackquoteclose\ and \isacharbackquoteopen@{text "B |
|
766 y"}\isacharbackquoteclose\ are referenced immediately; there is no |
|
767 need to decompose the logical rule structure again. In the second |
|
768 proof the final ``@{command then}~@{command show}~@{text |
|
769 thesis}~@{command ".."}'' involves the local rule case @{text "\<And>x |
|
770 y. A x \<Longrightarrow> B y \<Longrightarrow> thesis"} for the particular instance of terms @{text |
|
771 "a"} and @{text "b"} produced in the body. |
|
772 *} |
|
773 |
|
774 |
|
775 subsection {* Structured proof refinement \label{sec:framework-subproof} *} |
|
776 |
|
777 text {* |
|
778 By breaking up the grammar for the Isar proof language, we may |
|
779 understand a proof text as a linear sequence of individual proof |
|
780 commands. These are interpreted as transitions of the Isar virtual |
|
781 machine (Isar/VM), which operates on a block-structured |
|
782 configuration in single steps. This allows users to write proof |
|
783 texts in an incremental manner, and inspect intermediate |
|
784 configurations for debugging. |
|
785 |
|
786 The basic idea is analogous to evaluating algebraic expressions on a |
|
787 stack machine: @{text "(a + b) \<cdot> c"} then corresponds to a sequence |
|
788 of single transitions for each symbol @{text "(, a, +, b, ), \<cdot>, c"}. |
|
789 In Isar the algebraic values are facts or goals, and the operations |
|
790 are inferences. |
|
791 |
|
792 \medskip The Isar/VM state maintains a stack of nodes, each node |
|
793 contains the local proof context, the linguistic mode, and a pending |
|
794 goal (optional). The mode determines the type of transition that |
|
795 may be performed next, it essentially alternates between forward and |
|
796 backward reasoning, with an intermediate stage for chained facts |
|
797 (see \figref{fig:isar-vm}). |
|
798 |
|
799 \begin{figure}[htb] |
|
800 \begin{center} |
|
801 \includegraphics[width=0.8\textwidth]{isar-vm} |
|
802 \end{center} |
|
803 \caption{Isar/VM modes}\label{fig:isar-vm} |
|
804 \end{figure} |
|
805 |
|
806 For example, in @{text "state"} mode Isar acts like a mathematical |
|
807 scratch-pad, accepting declarations like @{command fix}, @{command |
|
808 assume}, and claims like @{command have}, @{command show}. A goal |
|
809 statement changes the mode to @{text "prove"}, which means that we |
|
810 may now refine the problem via @{command unfolding} or @{command |
|
811 proof}. Then we are again in @{text "state"} mode of a proof body, |
|
812 which may issue @{command show} statements to solve pending |
|
813 sub-goals. A concluding @{command qed} will return to the original |
|
814 @{text "state"} mode one level upwards. The subsequent Isar/VM |
|
815 trace indicates block structure, linguistic mode, goal state, and |
|
816 inferences: |
|
817 *} |
|
818 |
|
819 text_raw {* \begingroup\footnotesize *} |
|
820 (*<*)notepad begin |
|
821 (*>*) |
|
822 txt_raw {* \begin{minipage}[t]{0.18\textwidth} *} |
|
823 have "A \<longrightarrow> B" |
|
824 proof |
|
825 assume A |
|
826 show B |
|
827 sorry %noproof |
|
828 qed |
|
829 txt_raw {* \end{minipage}\quad |
|
830 \begin{minipage}[t]{0.06\textwidth} |
|
831 @{text "begin"} \\ |
|
832 \\ |
|
833 \\ |
|
834 @{text "begin"} \\ |
|
835 @{text "end"} \\ |
|
836 @{text "end"} \\ |
|
837 \end{minipage} |
|
838 \begin{minipage}[t]{0.08\textwidth} |
|
839 @{text "prove"} \\ |
|
840 @{text "state"} \\ |
|
841 @{text "state"} \\ |
|
842 @{text "prove"} \\ |
|
843 @{text "state"} \\ |
|
844 @{text "state"} \\ |
|
845 \end{minipage}\begin{minipage}[t]{0.35\textwidth} |
|
846 @{text "(A \<longrightarrow> B) \<Longrightarrow> #(A \<longrightarrow> B)"} \\ |
|
847 @{text "(A \<Longrightarrow> B) \<Longrightarrow> #(A \<longrightarrow> B)"} \\ |
|
848 \\ |
|
849 \\ |
|
850 @{text "#(A \<longrightarrow> B)"} \\ |
|
851 @{text "A \<longrightarrow> B"} \\ |
|
852 \end{minipage}\begin{minipage}[t]{0.4\textwidth} |
|
853 @{text "(init)"} \\ |
|
854 @{text "(resolution impI)"} \\ |
|
855 \\ |
|
856 \\ |
|
857 @{text "(refinement #A \<Longrightarrow> B)"} \\ |
|
858 @{text "(finish)"} \\ |
|
859 \end{minipage} *} |
|
860 (*<*) |
|
861 end |
|
862 (*>*) |
|
863 text_raw {* \endgroup *} |
|
864 |
|
865 text {* |
|
866 \noindent Here the @{inference refinement} inference from |
|
867 \secref{sec:framework-resolution} mediates composition of Isar |
|
868 sub-proofs nicely. Observe that this principle incorporates some |
|
869 degree of freedom in proof composition. In particular, the proof |
|
870 body allows parameters and assumptions to be re-ordered, or commuted |
|
871 according to Hereditary Harrop Form. Moreover, context elements |
|
872 that are not used in a sub-proof may be omitted altogether. For |
|
873 example: |
|
874 *} |
|
875 |
|
876 text_raw {*\begin{minipage}{0.5\textwidth}*} |
|
877 |
|
878 (*<*) |
|
879 notepad |
|
880 begin |
|
881 (*>*) |
|
882 have "\<And>x y. A x \<Longrightarrow> B y \<Longrightarrow> C x y" |
|
883 proof - |
|
884 fix x and y |
|
885 assume "A x" and "B y" |
|
886 show "C x y" sorry %noproof |
|
887 qed |
|
888 |
|
889 txt_raw {*\end{minipage}\begin{minipage}{0.5\textwidth}*} |
|
890 |
|
891 (*<*) |
|
892 next |
|
893 (*>*) |
|
894 have "\<And>x y. A x \<Longrightarrow> B y \<Longrightarrow> C x y" |
|
895 proof - |
|
896 fix x assume "A x" |
|
897 fix y assume "B y" |
|
898 show "C x y" sorry %noproof |
|
899 qed |
|
900 |
|
901 txt_raw {*\end{minipage}\\[3ex]\begin{minipage}{0.5\textwidth}*} |
|
902 |
|
903 (*<*) |
|
904 next |
|
905 (*>*) |
|
906 have "\<And>x y. A x \<Longrightarrow> B y \<Longrightarrow> C x y" |
|
907 proof - |
|
908 fix y assume "B y" |
|
909 fix x assume "A x" |
|
910 show "C x y" sorry |
|
911 qed |
|
912 |
|
913 txt_raw {*\end{minipage}\begin{minipage}{0.5\textwidth}*} |
|
914 (*<*) |
|
915 next |
|
916 (*>*) |
|
917 have "\<And>x y. A x \<Longrightarrow> B y \<Longrightarrow> C x y" |
|
918 proof - |
|
919 fix y assume "B y" |
|
920 fix x |
|
921 show "C x y" sorry |
|
922 qed |
|
923 (*<*) |
|
924 end |
|
925 (*>*) |
|
926 |
|
927 text_raw {*\end{minipage}*} |
|
928 |
|
929 text {* |
|
930 \medskip\noindent Such ``peephole optimizations'' of Isar texts are |
|
931 practically important to improve readability, by rearranging |
|
932 contexts elements according to the natural flow of reasoning in the |
|
933 body, while still observing the overall scoping rules. |
|
934 |
|
935 \medskip This illustrates the basic idea of structured proof |
|
936 processing in Isar. The main mechanisms are based on natural |
|
937 deduction rule composition within the Pure framework. In |
|
938 particular, there are no direct operations on goal states within the |
|
939 proof body. Moreover, there is no hidden automated reasoning |
|
940 involved, just plain unification. |
|
941 *} |
|
942 |
|
943 |
|
944 subsection {* Calculational reasoning \label{sec:framework-calc} *} |
|
945 |
|
946 text {* |
|
947 The existing Isar infrastructure is sufficiently flexible to support |
|
948 calculational reasoning (chains of transitivity steps) as derived |
|
949 concept. The generic proof elements introduced below depend on |
|
950 rules declared as @{attribute trans} in the context. It is left to |
|
951 the object-logic to provide a suitable rule collection for mixed |
|
952 relations of @{text "="}, @{text "<"}, @{text "\<le>"}, @{text "\<subset>"}, |
|
953 @{text "\<subseteq>"} etc. Due to the flexibility of rule composition |
|
954 (\secref{sec:framework-resolution}), substitution of equals by |
|
955 equals is covered as well, even substitution of inequalities |
|
956 involving monotonicity conditions; see also \cite[\S6]{Wenzel-PhD} |
|
957 and \cite{Bauer-Wenzel:2001}. |
|
958 |
|
959 The generic calculational mechanism is based on the observation that |
|
960 rules such as @{text "trans:"}~@{prop "x = y \<Longrightarrow> y = z \<Longrightarrow> x = z"} |
|
961 proceed from the premises towards the conclusion in a deterministic |
|
962 fashion. Thus we may reason in forward mode, feeding intermediate |
|
963 results into rules selected from the context. The course of |
|
964 reasoning is organized by maintaining a secondary fact called |
|
965 ``@{fact calculation}'', apart from the primary ``@{fact this}'' |
|
966 already provided by the Isar primitives. In the definitions below, |
|
967 @{attribute OF} refers to @{inference resolution} |
|
968 (\secref{sec:framework-resolution}) with multiple rule arguments, |
|
969 and @{text "trans"} represents to a suitable rule from the context: |
|
970 |
|
971 \begin{matharray}{rcl} |
|
972 @{command "also"}@{text "\<^sub>0"} & \equiv & @{command "note"}~@{text "calculation = this"} \\ |
|
973 @{command "also"}@{text "\<^sub>n\<^sub>+\<^sub>1"} & \equiv & @{command "note"}~@{text "calculation = trans [OF calculation this]"} \\[0.5ex] |
|
974 @{command "finally"} & \equiv & @{command "also"}~@{command "from"}~@{text calculation} \\ |
|
975 \end{matharray} |
|
976 |
|
977 \noindent The start of a calculation is determined implicitly in the |
|
978 text: here @{command also} sets @{fact calculation} to the current |
|
979 result; any subsequent occurrence will update @{fact calculation} by |
|
980 combination with the next result and a transitivity rule. The |
|
981 calculational sequence is concluded via @{command finally}, where |
|
982 the final result is exposed for use in a concluding claim. |
|
983 |
|
984 Here is a canonical proof pattern, using @{command have} to |
|
985 establish the intermediate results: |
|
986 *} |
|
987 |
|
988 (*<*) |
|
989 notepad |
|
990 begin |
|
991 (*>*) |
|
992 have "a = b" sorry |
|
993 also have "\<dots> = c" sorry |
|
994 also have "\<dots> = d" sorry |
|
995 finally have "a = d" . |
|
996 (*<*) |
|
997 end |
|
998 (*>*) |
|
999 |
|
1000 text {* |
|
1001 \noindent The term ``@{text "\<dots>"}'' above is a special abbreviation |
|
1002 provided by the Isabelle/Isar syntax layer: it statically refers to |
|
1003 the right-hand side argument of the previous statement given in the |
|
1004 text. Thus it happens to coincide with relevant sub-expressions in |
|
1005 the calculational chain, but the exact correspondence is dependent |
|
1006 on the transitivity rules being involved. |
|
1007 |
|
1008 \medskip Symmetry rules such as @{prop "x = y \<Longrightarrow> y = x"} are like |
|
1009 transitivities with only one premise. Isar maintains a separate |
|
1010 rule collection declared via the @{attribute sym} attribute, to be |
|
1011 used in fact expressions ``@{text "a [symmetric]"}'', or single-step |
|
1012 proofs ``@{command assume}~@{text "x = y"}~@{command then}~@{command |
|
1013 have}~@{text "y = x"}~@{command ".."}''. |
|
1014 *} |
|
1015 |
|
1016 end |