1 theory HOL_Specific |
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2 imports Base Main "~~/src/HOL/Library/Old_Recdef" |
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3 begin |
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4 |
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5 chapter {* Isabelle/HOL \label{ch:hol} *} |
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6 |
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7 section {* Higher-Order Logic *} |
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8 |
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9 text {* Isabelle/HOL is based on Higher-Order Logic, a polymorphic |
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10 version of Church's Simple Theory of Types. HOL can be best |
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11 understood as a simply-typed version of classical set theory. The |
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12 logic was first implemented in Gordon's HOL system |
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13 \cite{mgordon-hol}. It extends Church's original logic |
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14 \cite{church40} by explicit type variables (naive polymorphism) and |
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15 a sound axiomatization scheme for new types based on subsets of |
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16 existing types. |
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17 |
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18 Andrews's book \cite{andrews86} is a full description of the |
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19 original Church-style higher-order logic, with proofs of correctness |
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20 and completeness wrt.\ certain set-theoretic interpretations. The |
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21 particular extensions of Gordon-style HOL are explained semantically |
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22 in two chapters of the 1993 HOL book \cite{pitts93}. |
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23 |
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24 Experience with HOL over decades has demonstrated that higher-order |
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25 logic is widely applicable in many areas of mathematics and computer |
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26 science. In a sense, Higher-Order Logic is simpler than First-Order |
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27 Logic, because there are fewer restrictions and special cases. Note |
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28 that HOL is \emph{weaker} than FOL with axioms for ZF set theory, |
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29 which is traditionally considered the standard foundation of regular |
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30 mathematics, but for most applications this does not matter. If you |
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31 prefer ML to Lisp, you will probably prefer HOL to ZF. |
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32 |
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33 \medskip The syntax of HOL follows @{text "\<lambda>"}-calculus and |
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34 functional programming. Function application is curried. To apply |
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35 the function @{text f} of type @{text "\<tau>\<^sub>1 \<Rightarrow> \<tau>\<^sub>2 \<Rightarrow> \<tau>\<^sub>3"} to the |
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36 arguments @{text a} and @{text b} in HOL, you simply write @{text "f |
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37 a b"} (as in ML or Haskell). There is no ``apply'' operator; the |
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38 existing application of the Pure @{text "\<lambda>"}-calculus is re-used. |
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39 Note that in HOL @{text "f (a, b)"} means ``@{text "f"} applied to |
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40 the pair @{text "(a, b)"} (which is notation for @{text "Pair a |
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41 b"}). The latter typically introduces extra formal efforts that can |
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42 be avoided by currying functions by default. Explicit tuples are as |
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43 infrequent in HOL formalizations as in good ML or Haskell programs. |
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44 |
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45 \medskip Isabelle/HOL has a distinct feel, compared to other |
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46 object-logics like Isabelle/ZF. It identifies object-level types |
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47 with meta-level types, taking advantage of the default |
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48 type-inference mechanism of Isabelle/Pure. HOL fully identifies |
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49 object-level functions with meta-level functions, with native |
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50 abstraction and application. |
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51 |
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52 These identifications allow Isabelle to support HOL particularly |
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53 nicely, but they also mean that HOL requires some sophistication |
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54 from the user. In particular, an understanding of Hindley-Milner |
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55 type-inference with type-classes, which are both used extensively in |
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56 the standard libraries and applications. Beginners can set |
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57 @{attribute show_types} or even @{attribute show_sorts} to get more |
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58 explicit information about the result of type-inference. *} |
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59 |
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60 |
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61 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *} |
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62 |
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63 text {* |
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64 \begin{matharray}{rcl} |
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65 @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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66 @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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67 @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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68 @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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69 @{attribute_def (HOL) mono} & : & @{text attribute} \\ |
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70 \end{matharray} |
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71 |
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72 An \emph{inductive definition} specifies the least predicate or set |
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73 @{text R} closed under given rules: applying a rule to elements of |
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74 @{text R} yields a result within @{text R}. For example, a |
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75 structural operational semantics is an inductive definition of an |
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76 evaluation relation. |
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77 |
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78 Dually, a \emph{coinductive definition} specifies the greatest |
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79 predicate or set @{text R} that is consistent with given rules: |
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80 every element of @{text R} can be seen as arising by applying a rule |
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81 to elements of @{text R}. An important example is using |
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82 bisimulation relations to formalise equivalence of processes and |
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83 infinite data structures. |
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84 |
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85 Both inductive and coinductive definitions are based on the |
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86 Knaster-Tarski fixed-point theorem for complete lattices. The |
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87 collection of introduction rules given by the user determines a |
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88 functor on subsets of set-theoretic relations. The required |
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89 monotonicity of the recursion scheme is proven as a prerequisite to |
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90 the fixed-point definition and the resulting consequences. This |
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91 works by pushing inclusion through logical connectives and any other |
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92 operator that might be wrapped around recursive occurrences of the |
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93 defined relation: there must be a monotonicity theorem of the form |
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94 @{text "A \<le> B \<Longrightarrow> \<M> A \<le> \<M> B"}, for each premise @{text "\<M> R t"} in an |
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95 introduction rule. The default rule declarations of Isabelle/HOL |
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96 already take care of most common situations. |
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97 |
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98 @{rail " |
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99 (@@{command (HOL) inductive} | @@{command (HOL) inductive_set} | |
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100 @@{command (HOL) coinductive} | @@{command (HOL) coinductive_set}) |
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101 @{syntax target}? \\ |
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102 @{syntax \"fixes\"} (@'for' @{syntax \"fixes\"})? (@'where' clauses)? \\ |
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103 (@'monos' @{syntax thmrefs})? |
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104 ; |
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105 clauses: (@{syntax thmdecl}? @{syntax prop} + '|') |
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106 ; |
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107 @@{attribute (HOL) mono} (() | 'add' | 'del') |
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108 "} |
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109 |
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110 \begin{description} |
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111 |
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112 \item @{command (HOL) "inductive"} and @{command (HOL) |
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113 "coinductive"} define (co)inductive predicates from the introduction |
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114 rules. |
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115 |
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116 The propositions given as @{text "clauses"} in the @{keyword |
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117 "where"} part are either rules of the usual @{text "\<And>/\<Longrightarrow>"} format |
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118 (with arbitrary nesting), or equalities using @{text "\<equiv>"}. The |
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119 latter specifies extra-logical abbreviations in the sense of |
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120 @{command_ref abbreviation}. Introducing abstract syntax |
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121 simultaneously with the actual introduction rules is occasionally |
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122 useful for complex specifications. |
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123 |
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124 The optional @{keyword "for"} part contains a list of parameters of |
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125 the (co)inductive predicates that remain fixed throughout the |
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126 definition, in contrast to arguments of the relation that may vary |
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127 in each occurrence within the given @{text "clauses"}. |
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128 |
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129 The optional @{keyword "monos"} declaration contains additional |
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130 \emph{monotonicity theorems}, which are required for each operator |
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131 applied to a recursive set in the introduction rules. |
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132 |
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133 \item @{command (HOL) "inductive_set"} and @{command (HOL) |
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134 "coinductive_set"} are wrappers for to the previous commands for |
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135 native HOL predicates. This allows to define (co)inductive sets, |
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136 where multiple arguments are simulated via tuples. |
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137 |
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138 \item @{attribute (HOL) mono} declares monotonicity rules in the |
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139 context. These rule are involved in the automated monotonicity |
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140 proof of the above inductive and coinductive definitions. |
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141 |
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142 \end{description} |
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143 *} |
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144 |
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145 |
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146 subsection {* Derived rules *} |
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147 |
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148 text {* A (co)inductive definition of @{text R} provides the following |
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149 main theorems: |
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150 |
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151 \begin{description} |
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152 |
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153 \item @{text R.intros} is the list of introduction rules as proven |
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154 theorems, for the recursive predicates (or sets). The rules are |
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155 also available individually, using the names given them in the |
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156 theory file; |
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157 |
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158 \item @{text R.cases} is the case analysis (or elimination) rule; |
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159 |
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160 \item @{text R.induct} or @{text R.coinduct} is the (co)induction |
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161 rule; |
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162 |
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163 \item @{text R.simps} is the equation unrolling the fixpoint of the |
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164 predicate one step. |
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165 |
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166 \end{description} |
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167 |
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168 When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are |
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169 defined simultaneously, the list of introduction rules is called |
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170 @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are |
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171 called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list |
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172 of mutual induction rules is called @{text |
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173 "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}. |
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174 *} |
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175 |
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176 |
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177 subsection {* Monotonicity theorems *} |
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178 |
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179 text {* The context maintains a default set of theorems that are used |
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180 in monotonicity proofs. New rules can be declared via the |
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181 @{attribute (HOL) mono} attribute. See the main Isabelle/HOL |
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182 sources for some examples. The general format of such monotonicity |
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183 theorems is as follows: |
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184 |
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185 \begin{itemize} |
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186 |
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187 \item Theorems of the form @{text "A \<le> B \<Longrightarrow> \<M> A \<le> \<M> B"}, for proving |
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188 monotonicity of inductive definitions whose introduction rules have |
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189 premises involving terms such as @{text "\<M> R t"}. |
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190 |
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191 \item Monotonicity theorems for logical operators, which are of the |
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192 general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}. For example, in |
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193 the case of the operator @{text "\<or>"}, the corresponding theorem is |
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194 \[ |
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195 \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}} |
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196 \] |
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197 |
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198 \item De Morgan style equations for reasoning about the ``polarity'' |
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199 of expressions, e.g. |
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200 \[ |
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201 @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad |
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202 @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"} |
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203 \] |
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204 |
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205 \item Equations for reducing complex operators to more primitive |
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206 ones whose monotonicity can easily be proved, e.g. |
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207 \[ |
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208 @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad |
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209 @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"} |
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210 \] |
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211 |
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212 \end{itemize} |
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213 *} |
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214 |
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215 subsubsection {* Examples *} |
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216 |
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217 text {* The finite powerset operator can be defined inductively like this: *} |
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218 |
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219 inductive_set Fin :: "'a set \<Rightarrow> 'a set set" for A :: "'a set" |
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220 where |
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221 empty: "{} \<in> Fin A" |
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222 | insert: "a \<in> A \<Longrightarrow> B \<in> Fin A \<Longrightarrow> insert a B \<in> Fin A" |
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223 |
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224 text {* The accessible part of a relation is defined as follows: *} |
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225 |
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226 inductive acc :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" |
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227 for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<prec>" 50) |
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228 where acc: "(\<And>y. y \<prec> x \<Longrightarrow> acc r y) \<Longrightarrow> acc r x" |
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229 |
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230 text {* Common logical connectives can be easily characterized as |
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231 non-recursive inductive definitions with parameters, but without |
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232 arguments. *} |
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233 |
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234 inductive AND for A B :: bool |
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235 where "A \<Longrightarrow> B \<Longrightarrow> AND A B" |
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236 |
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237 inductive OR for A B :: bool |
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238 where "A \<Longrightarrow> OR A B" |
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239 | "B \<Longrightarrow> OR A B" |
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240 |
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241 inductive EXISTS for B :: "'a \<Rightarrow> bool" |
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242 where "B a \<Longrightarrow> EXISTS B" |
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243 |
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244 text {* Here the @{text "cases"} or @{text "induct"} rules produced by |
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245 the @{command inductive} package coincide with the expected |
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246 elimination rules for Natural Deduction. Already in the original |
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247 article by Gerhard Gentzen \cite{Gentzen:1935} there is a hint that |
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248 each connective can be characterized by its introductions, and the |
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249 elimination can be constructed systematically. *} |
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250 |
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251 |
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252 section {* Recursive functions \label{sec:recursion} *} |
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253 |
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254 text {* |
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255 \begin{matharray}{rcl} |
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256 @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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257 @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
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258 @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\ |
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259 @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\ |
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260 \end{matharray} |
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261 |
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262 @{rail " |
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263 @@{command (HOL) primrec} @{syntax target}? @{syntax \"fixes\"} @'where' equations |
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264 ; |
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265 (@@{command (HOL) fun} | @@{command (HOL) function}) @{syntax target}? functionopts? |
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266 @{syntax \"fixes\"} \\ @'where' equations |
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267 ; |
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268 |
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269 equations: (@{syntax thmdecl}? @{syntax prop} + '|') |
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270 ; |
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271 functionopts: '(' (('sequential' | 'domintros') + ',') ')' |
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272 ; |
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273 @@{command (HOL) termination} @{syntax term}? |
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274 "} |
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275 |
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276 \begin{description} |
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277 |
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278 \item @{command (HOL) "primrec"} defines primitive recursive |
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279 functions over datatypes (see also @{command_ref (HOL) datatype} and |
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280 @{command_ref (HOL) rep_datatype}). The given @{text equations} |
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281 specify reduction rules that are produced by instantiating the |
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282 generic combinator for primitive recursion that is available for |
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283 each datatype. |
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284 |
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285 Each equation needs to be of the form: |
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286 |
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287 @{text [display] "f x\<^sub>1 \<dots> x\<^sub>m (C y\<^sub>1 \<dots> y\<^sub>k) z\<^sub>1 \<dots> z\<^sub>n = rhs"} |
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288 |
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289 such that @{text C} is a datatype constructor, @{text rhs} contains |
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290 only the free variables on the left-hand side (or from the context), |
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291 and all recursive occurrences of @{text "f"} in @{text "rhs"} are of |
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292 the form @{text "f \<dots> y\<^sub>i \<dots>"} for some @{text i}. At most one |
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293 reduction rule for each constructor can be given. The order does |
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294 not matter. For missing constructors, the function is defined to |
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295 return a default value, but this equation is made difficult to |
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296 access for users. |
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297 |
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298 The reduction rules are declared as @{attribute simp} by default, |
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299 which enables standard proof methods like @{method simp} and |
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300 @{method auto} to normalize expressions of @{text "f"} applied to |
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301 datatype constructions, by simulating symbolic computation via |
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302 rewriting. |
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303 |
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304 \item @{command (HOL) "function"} defines functions by general |
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305 wellfounded recursion. A detailed description with examples can be |
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306 found in \cite{isabelle-function}. The function is specified by a |
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307 set of (possibly conditional) recursive equations with arbitrary |
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308 pattern matching. The command generates proof obligations for the |
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309 completeness and the compatibility of patterns. |
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310 |
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311 The defined function is considered partial, and the resulting |
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312 simplification rules (named @{text "f.psimps"}) and induction rule |
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313 (named @{text "f.pinduct"}) are guarded by a generated domain |
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314 predicate @{text "f_dom"}. The @{command (HOL) "termination"} |
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315 command can then be used to establish that the function is total. |
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316 |
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317 \item @{command (HOL) "fun"} is a shorthand notation for ``@{command |
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318 (HOL) "function"}~@{text "(sequential)"}, followed by automated |
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319 proof attempts regarding pattern matching and termination. See |
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320 \cite{isabelle-function} for further details. |
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321 |
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322 \item @{command (HOL) "termination"}~@{text f} commences a |
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323 termination proof for the previously defined function @{text f}. If |
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324 this is omitted, the command refers to the most recent function |
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325 definition. After the proof is closed, the recursive equations and |
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326 the induction principle is established. |
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327 |
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328 \end{description} |
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329 |
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330 Recursive definitions introduced by the @{command (HOL) "function"} |
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331 command accommodate reasoning by induction (cf.\ @{method induct}): |
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332 rule @{text "f.induct"} refers to a specific induction rule, with |
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333 parameters named according to the user-specified equations. Cases |
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334 are numbered starting from 1. For @{command (HOL) "primrec"}, the |
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335 induction principle coincides with structural recursion on the |
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336 datatype where the recursion is carried out. |
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337 |
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338 The equations provided by these packages may be referred later as |
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339 theorem list @{text "f.simps"}, where @{text f} is the (collective) |
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340 name of the functions defined. Individual equations may be named |
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341 explicitly as well. |
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342 |
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343 The @{command (HOL) "function"} command accepts the following |
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344 options. |
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345 |
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346 \begin{description} |
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347 |
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348 \item @{text sequential} enables a preprocessor which disambiguates |
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349 overlapping patterns by making them mutually disjoint. Earlier |
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350 equations take precedence over later ones. This allows to give the |
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351 specification in a format very similar to functional programming. |
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352 Note that the resulting simplification and induction rules |
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353 correspond to the transformed specification, not the one given |
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354 originally. This usually means that each equation given by the user |
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355 may result in several theorems. Also note that this automatic |
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356 transformation only works for ML-style datatype patterns. |
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357 |
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358 \item @{text domintros} enables the automated generation of |
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359 introduction rules for the domain predicate. While mostly not |
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360 needed, they can be helpful in some proofs about partial functions. |
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361 |
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362 \end{description} |
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363 *} |
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364 |
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365 subsubsection {* Example: evaluation of expressions *} |
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366 |
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367 text {* Subsequently, we define mutual datatypes for arithmetic and |
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368 boolean expressions, and use @{command primrec} for evaluation |
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369 functions that follow the same recursive structure. *} |
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370 |
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371 datatype 'a aexp = |
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372 IF "'a bexp" "'a aexp" "'a aexp" |
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373 | Sum "'a aexp" "'a aexp" |
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374 | Diff "'a aexp" "'a aexp" |
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375 | Var 'a |
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376 | Num nat |
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377 and 'a bexp = |
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378 Less "'a aexp" "'a aexp" |
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379 | And "'a bexp" "'a bexp" |
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380 | Neg "'a bexp" |
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381 |
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382 |
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383 text {* \medskip Evaluation of arithmetic and boolean expressions *} |
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384 |
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385 primrec evala :: "('a \<Rightarrow> nat) \<Rightarrow> 'a aexp \<Rightarrow> nat" |
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386 and evalb :: "('a \<Rightarrow> nat) \<Rightarrow> 'a bexp \<Rightarrow> bool" |
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387 where |
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388 "evala env (IF b a1 a2) = (if evalb env b then evala env a1 else evala env a2)" |
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389 | "evala env (Sum a1 a2) = evala env a1 + evala env a2" |
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390 | "evala env (Diff a1 a2) = evala env a1 - evala env a2" |
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391 | "evala env (Var v) = env v" |
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392 | "evala env (Num n) = n" |
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393 | "evalb env (Less a1 a2) = (evala env a1 < evala env a2)" |
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394 | "evalb env (And b1 b2) = (evalb env b1 \<and> evalb env b2)" |
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395 | "evalb env (Neg b) = (\<not> evalb env b)" |
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396 |
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397 text {* Since the value of an expression depends on the value of its |
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398 variables, the functions @{const evala} and @{const evalb} take an |
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399 additional parameter, an \emph{environment} that maps variables to |
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400 their values. |
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401 |
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402 \medskip Substitution on expressions can be defined similarly. The |
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403 mapping @{text f} of type @{typ "'a \<Rightarrow> 'a aexp"} given as a |
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404 parameter is lifted canonically on the types @{typ "'a aexp"} and |
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405 @{typ "'a bexp"}, respectively. |
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406 *} |
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407 |
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408 primrec substa :: "('a \<Rightarrow> 'b aexp) \<Rightarrow> 'a aexp \<Rightarrow> 'b aexp" |
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409 and substb :: "('a \<Rightarrow> 'b aexp) \<Rightarrow> 'a bexp \<Rightarrow> 'b bexp" |
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410 where |
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411 "substa f (IF b a1 a2) = IF (substb f b) (substa f a1) (substa f a2)" |
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412 | "substa f (Sum a1 a2) = Sum (substa f a1) (substa f a2)" |
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413 | "substa f (Diff a1 a2) = Diff (substa f a1) (substa f a2)" |
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414 | "substa f (Var v) = f v" |
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415 | "substa f (Num n) = Num n" |
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416 | "substb f (Less a1 a2) = Less (substa f a1) (substa f a2)" |
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417 | "substb f (And b1 b2) = And (substb f b1) (substb f b2)" |
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418 | "substb f (Neg b) = Neg (substb f b)" |
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419 |
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420 text {* In textbooks about semantics one often finds substitution |
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421 theorems, which express the relationship between substitution and |
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422 evaluation. For @{typ "'a aexp"} and @{typ "'a bexp"}, we can prove |
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423 such a theorem by mutual induction, followed by simplification. |
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424 *} |
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425 |
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426 lemma subst_one: |
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427 "evala env (substa (Var (v := a')) a) = evala (env (v := evala env a')) a" |
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428 "evalb env (substb (Var (v := a')) b) = evalb (env (v := evala env a')) b" |
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429 by (induct a and b) simp_all |
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430 |
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431 lemma subst_all: |
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432 "evala env (substa s a) = evala (\<lambda>x. evala env (s x)) a" |
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433 "evalb env (substb s b) = evalb (\<lambda>x. evala env (s x)) b" |
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434 by (induct a and b) simp_all |
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435 |
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436 |
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437 subsubsection {* Example: a substitution function for terms *} |
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438 |
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439 text {* Functions on datatypes with nested recursion are also defined |
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440 by mutual primitive recursion. *} |
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441 |
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442 datatype ('a, 'b) "term" = Var 'a | App 'b "('a, 'b) term list" |
|
443 |
|
444 text {* A substitution function on type @{typ "('a, 'b) term"} can be |
|
445 defined as follows, by working simultaneously on @{typ "('a, 'b) |
|
446 term list"}: *} |
|
447 |
|
448 primrec subst_term :: "('a \<Rightarrow> ('a, 'b) term) \<Rightarrow> ('a, 'b) term \<Rightarrow> ('a, 'b) term" and |
|
449 subst_term_list :: "('a \<Rightarrow> ('a, 'b) term) \<Rightarrow> ('a, 'b) term list \<Rightarrow> ('a, 'b) term list" |
|
450 where |
|
451 "subst_term f (Var a) = f a" |
|
452 | "subst_term f (App b ts) = App b (subst_term_list f ts)" |
|
453 | "subst_term_list f [] = []" |
|
454 | "subst_term_list f (t # ts) = subst_term f t # subst_term_list f ts" |
|
455 |
|
456 text {* The recursion scheme follows the structure of the unfolded |
|
457 definition of type @{typ "('a, 'b) term"}. To prove properties of this |
|
458 substitution function, mutual induction is needed: |
|
459 *} |
|
460 |
|
461 lemma "subst_term (subst_term f1 \<circ> f2) t = subst_term f1 (subst_term f2 t)" and |
|
462 "subst_term_list (subst_term f1 \<circ> f2) ts = subst_term_list f1 (subst_term_list f2 ts)" |
|
463 by (induct t and ts) simp_all |
|
464 |
|
465 |
|
466 subsubsection {* Example: a map function for infinitely branching trees *} |
|
467 |
|
468 text {* Defining functions on infinitely branching datatypes by |
|
469 primitive recursion is just as easy. |
|
470 *} |
|
471 |
|
472 datatype 'a tree = Atom 'a | Branch "nat \<Rightarrow> 'a tree" |
|
473 |
|
474 primrec map_tree :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a tree \<Rightarrow> 'b tree" |
|
475 where |
|
476 "map_tree f (Atom a) = Atom (f a)" |
|
477 | "map_tree f (Branch ts) = Branch (\<lambda>x. map_tree f (ts x))" |
|
478 |
|
479 text {* Note that all occurrences of functions such as @{text ts} |
|
480 above must be applied to an argument. In particular, @{term |
|
481 "map_tree f \<circ> ts"} is not allowed here. *} |
|
482 |
|
483 text {* Here is a simple composition lemma for @{term map_tree}: *} |
|
484 |
|
485 lemma "map_tree g (map_tree f t) = map_tree (g \<circ> f) t" |
|
486 by (induct t) simp_all |
|
487 |
|
488 |
|
489 subsection {* Proof methods related to recursive definitions *} |
|
490 |
|
491 text {* |
|
492 \begin{matharray}{rcl} |
|
493 @{method_def (HOL) pat_completeness} & : & @{text method} \\ |
|
494 @{method_def (HOL) relation} & : & @{text method} \\ |
|
495 @{method_def (HOL) lexicographic_order} & : & @{text method} \\ |
|
496 @{method_def (HOL) size_change} & : & @{text method} \\ |
|
497 @{method_def (HOL) induction_schema} & : & @{text method} \\ |
|
498 \end{matharray} |
|
499 |
|
500 @{rail " |
|
501 @@{method (HOL) relation} @{syntax term} |
|
502 ; |
|
503 @@{method (HOL) lexicographic_order} (@{syntax clasimpmod} * ) |
|
504 ; |
|
505 @@{method (HOL) size_change} ( orders (@{syntax clasimpmod} * ) ) |
|
506 ; |
|
507 @@{method (HOL) induction_schema} |
|
508 ; |
|
509 orders: ( 'max' | 'min' | 'ms' ) * |
|
510 "} |
|
511 |
|
512 \begin{description} |
|
513 |
|
514 \item @{method (HOL) pat_completeness} is a specialized method to |
|
515 solve goals regarding the completeness of pattern matching, as |
|
516 required by the @{command (HOL) "function"} package (cf.\ |
|
517 \cite{isabelle-function}). |
|
518 |
|
519 \item @{method (HOL) relation}~@{text R} introduces a termination |
|
520 proof using the relation @{text R}. The resulting proof state will |
|
521 contain goals expressing that @{text R} is wellfounded, and that the |
|
522 arguments of recursive calls decrease with respect to @{text R}. |
|
523 Usually, this method is used as the initial proof step of manual |
|
524 termination proofs. |
|
525 |
|
526 \item @{method (HOL) "lexicographic_order"} attempts a fully |
|
527 automated termination proof by searching for a lexicographic |
|
528 combination of size measures on the arguments of the function. The |
|
529 method accepts the same arguments as the @{method auto} method, |
|
530 which it uses internally to prove local descents. The @{syntax |
|
531 clasimpmod} modifiers are accepted (as for @{method auto}). |
|
532 |
|
533 In case of failure, extensive information is printed, which can help |
|
534 to analyse the situation (cf.\ \cite{isabelle-function}). |
|
535 |
|
536 \item @{method (HOL) "size_change"} also works on termination goals, |
|
537 using a variation of the size-change principle, together with a |
|
538 graph decomposition technique (see \cite{krauss_phd} for details). |
|
539 Three kinds of orders are used internally: @{text max}, @{text min}, |
|
540 and @{text ms} (multiset), which is only available when the theory |
|
541 @{text Multiset} is loaded. When no order kinds are given, they are |
|
542 tried in order. The search for a termination proof uses SAT solving |
|
543 internally. |
|
544 |
|
545 For local descent proofs, the @{syntax clasimpmod} modifiers are |
|
546 accepted (as for @{method auto}). |
|
547 |
|
548 \item @{method (HOL) induction_schema} derives user-specified |
|
549 induction rules from well-founded induction and completeness of |
|
550 patterns. This factors out some operations that are done internally |
|
551 by the function package and makes them available separately. See |
|
552 @{file "~~/src/HOL/ex/Induction_Schema.thy"} for examples. |
|
553 |
|
554 \end{description} |
|
555 *} |
|
556 |
|
557 |
|
558 subsection {* Functions with explicit partiality *} |
|
559 |
|
560 text {* |
|
561 \begin{matharray}{rcl} |
|
562 @{command_def (HOL) "partial_function"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
|
563 @{attribute_def (HOL) "partial_function_mono"} & : & @{text attribute} \\ |
|
564 \end{matharray} |
|
565 |
|
566 @{rail " |
|
567 @@{command (HOL) partial_function} @{syntax target}? |
|
568 '(' @{syntax nameref} ')' @{syntax \"fixes\"} \\ |
|
569 @'where' @{syntax thmdecl}? @{syntax prop} |
|
570 "} |
|
571 |
|
572 \begin{description} |
|
573 |
|
574 \item @{command (HOL) "partial_function"}~@{text "(mode)"} defines |
|
575 recursive functions based on fixpoints in complete partial |
|
576 orders. No termination proof is required from the user or |
|
577 constructed internally. Instead, the possibility of non-termination |
|
578 is modelled explicitly in the result type, which contains an |
|
579 explicit bottom element. |
|
580 |
|
581 Pattern matching and mutual recursion are currently not supported. |
|
582 Thus, the specification consists of a single function described by a |
|
583 single recursive equation. |
|
584 |
|
585 There are no fixed syntactic restrictions on the body of the |
|
586 function, but the induced functional must be provably monotonic |
|
587 wrt.\ the underlying order. The monotonicitity proof is performed |
|
588 internally, and the definition is rejected when it fails. The proof |
|
589 can be influenced by declaring hints using the |
|
590 @{attribute (HOL) partial_function_mono} attribute. |
|
591 |
|
592 The mandatory @{text mode} argument specifies the mode of operation |
|
593 of the command, which directly corresponds to a complete partial |
|
594 order on the result type. By default, the following modes are |
|
595 defined: |
|
596 |
|
597 \begin{description} |
|
598 |
|
599 \item @{text option} defines functions that map into the @{type |
|
600 option} type. Here, the value @{term None} is used to model a |
|
601 non-terminating computation. Monotonicity requires that if @{term |
|
602 None} is returned by a recursive call, then the overall result must |
|
603 also be @{term None}. This is best achieved through the use of the |
|
604 monadic operator @{const "Option.bind"}. |
|
605 |
|
606 \item @{text tailrec} defines functions with an arbitrary result |
|
607 type and uses the slightly degenerated partial order where @{term |
|
608 "undefined"} is the bottom element. Now, monotonicity requires that |
|
609 if @{term undefined} is returned by a recursive call, then the |
|
610 overall result must also be @{term undefined}. In practice, this is |
|
611 only satisfied when each recursive call is a tail call, whose result |
|
612 is directly returned. Thus, this mode of operation allows the |
|
613 definition of arbitrary tail-recursive functions. |
|
614 |
|
615 \end{description} |
|
616 |
|
617 Experienced users may define new modes by instantiating the locale |
|
618 @{const "partial_function_definitions"} appropriately. |
|
619 |
|
620 \item @{attribute (HOL) partial_function_mono} declares rules for |
|
621 use in the internal monononicity proofs of partial function |
|
622 definitions. |
|
623 |
|
624 \end{description} |
|
625 |
|
626 *} |
|
627 |
|
628 |
|
629 subsection {* Old-style recursive function definitions (TFL) *} |
|
630 |
|
631 text {* |
|
632 \begin{matharray}{rcl} |
|
633 @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\ |
|
634 @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ |
|
635 \end{matharray} |
|
636 |
|
637 The old TFL commands @{command (HOL) "recdef"} and @{command (HOL) |
|
638 "recdef_tc"} for defining recursive are mostly obsolete; @{command |
|
639 (HOL) "function"} or @{command (HOL) "fun"} should be used instead. |
|
640 |
|
641 @{rail " |
|
642 @@{command (HOL) recdef} ('(' @'permissive' ')')? \\ |
|
643 @{syntax name} @{syntax term} (@{syntax prop} +) hints? |
|
644 ; |
|
645 recdeftc @{syntax thmdecl}? tc |
|
646 ; |
|
647 hints: '(' @'hints' ( recdefmod * ) ')' |
|
648 ; |
|
649 recdefmod: (('recdef_simp' | 'recdef_cong' | 'recdef_wf') |
|
650 (() | 'add' | 'del') ':' @{syntax thmrefs}) | @{syntax clasimpmod} |
|
651 ; |
|
652 tc: @{syntax nameref} ('(' @{syntax nat} ')')? |
|
653 "} |
|
654 |
|
655 \begin{description} |
|
656 |
|
657 \item @{command (HOL) "recdef"} defines general well-founded |
|
658 recursive functions (using the TFL package), see also |
|
659 \cite{isabelle-HOL}. The ``@{text "(permissive)"}'' option tells |
|
660 TFL to recover from failed proof attempts, returning unfinished |
|
661 results. The @{text recdef_simp}, @{text recdef_cong}, and @{text |
|
662 recdef_wf} hints refer to auxiliary rules to be used in the internal |
|
663 automated proof process of TFL. Additional @{syntax clasimpmod} |
|
664 declarations may be given to tune the context of the Simplifier |
|
665 (cf.\ \secref{sec:simplifier}) and Classical reasoner (cf.\ |
|
666 \secref{sec:classical}). |
|
667 |
|
668 \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the |
|
669 proof for leftover termination condition number @{text i} (default |
|
670 1) as generated by a @{command (HOL) "recdef"} definition of |
|
671 constant @{text c}. |
|
672 |
|
673 Note that in most cases, @{command (HOL) "recdef"} is able to finish |
|
674 its internal proofs without manual intervention. |
|
675 |
|
676 \end{description} |
|
677 |
|
678 \medskip Hints for @{command (HOL) "recdef"} may be also declared |
|
679 globally, using the following attributes. |
|
680 |
|
681 \begin{matharray}{rcl} |
|
682 @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\ |
|
683 @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\ |
|
684 @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\ |
|
685 \end{matharray} |
|
686 |
|
687 @{rail " |
|
688 (@@{attribute (HOL) recdef_simp} | @@{attribute (HOL) recdef_cong} | |
|
689 @@{attribute (HOL) recdef_wf}) (() | 'add' | 'del') |
|
690 "} |
|
691 *} |
|
692 |
|
693 |
|
694 section {* Datatypes \label{sec:hol-datatype} *} |
|
695 |
|
696 text {* |
|
697 \begin{matharray}{rcl} |
|
698 @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
699 @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ |
|
700 \end{matharray} |
|
701 |
|
702 @{rail " |
|
703 @@{command (HOL) datatype} (spec + @'and') |
|
704 ; |
|
705 @@{command (HOL) rep_datatype} ('(' (@{syntax name} +) ')')? (@{syntax term} +) |
|
706 ; |
|
707 |
|
708 spec: @{syntax typespec_sorts} @{syntax mixfix}? '=' (cons + '|') |
|
709 ; |
|
710 cons: @{syntax name} (@{syntax type} * ) @{syntax mixfix}? |
|
711 "} |
|
712 |
|
713 \begin{description} |
|
714 |
|
715 \item @{command (HOL) "datatype"} defines inductive datatypes in |
|
716 HOL. |
|
717 |
|
718 \item @{command (HOL) "rep_datatype"} represents existing types as |
|
719 datatypes. |
|
720 |
|
721 For foundational reasons, some basic types such as @{typ nat}, @{typ |
|
722 "'a \<times> 'b"}, @{typ "'a + 'b"}, @{typ bool} and @{typ unit} are |
|
723 introduced by more primitive means using @{command_ref typedef}. To |
|
724 recover the rich infrastructure of @{command datatype} (e.g.\ rules |
|
725 for @{method cases} and @{method induct} and the primitive recursion |
|
726 combinators), such types may be represented as actual datatypes |
|
727 later. This is done by specifying the constructors of the desired |
|
728 type, and giving a proof of the induction rule, distinctness and |
|
729 injectivity of constructors. |
|
730 |
|
731 For example, see @{file "~~/src/HOL/Sum_Type.thy"} for the |
|
732 representation of the primitive sum type as fully-featured datatype. |
|
733 |
|
734 \end{description} |
|
735 |
|
736 The generated rules for @{method induct} and @{method cases} provide |
|
737 case names according to the given constructors, while parameters are |
|
738 named after the types (see also \secref{sec:cases-induct}). |
|
739 |
|
740 See \cite{isabelle-HOL} for more details on datatypes, but beware of |
|
741 the old-style theory syntax being used there! Apart from proper |
|
742 proof methods for case-analysis and induction, there are also |
|
743 emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL) |
|
744 induct_tac} available, see \secref{sec:hol-induct-tac}; these admit |
|
745 to refer directly to the internal structure of subgoals (including |
|
746 internally bound parameters). |
|
747 *} |
|
748 |
|
749 |
|
750 subsubsection {* Examples *} |
|
751 |
|
752 text {* We define a type of finite sequences, with slightly different |
|
753 names than the existing @{typ "'a list"} that is already in @{theory |
|
754 Main}: *} |
|
755 |
|
756 datatype 'a seq = Empty | Seq 'a "'a seq" |
|
757 |
|
758 text {* We can now prove some simple lemma by structural induction: *} |
|
759 |
|
760 lemma "Seq x xs \<noteq> xs" |
|
761 proof (induct xs arbitrary: x) |
|
762 case Empty |
|
763 txt {* This case can be proved using the simplifier: the freeness |
|
764 properties of the datatype are already declared as @{attribute |
|
765 simp} rules. *} |
|
766 show "Seq x Empty \<noteq> Empty" |
|
767 by simp |
|
768 next |
|
769 case (Seq y ys) |
|
770 txt {* The step case is proved similarly. *} |
|
771 show "Seq x (Seq y ys) \<noteq> Seq y ys" |
|
772 using `Seq y ys \<noteq> ys` by simp |
|
773 qed |
|
774 |
|
775 text {* Here is a more succinct version of the same proof: *} |
|
776 |
|
777 lemma "Seq x xs \<noteq> xs" |
|
778 by (induct xs arbitrary: x) simp_all |
|
779 |
|
780 |
|
781 section {* Records \label{sec:hol-record} *} |
|
782 |
|
783 text {* |
|
784 In principle, records merely generalize the concept of tuples, where |
|
785 components may be addressed by labels instead of just position. The |
|
786 logical infrastructure of records in Isabelle/HOL is slightly more |
|
787 advanced, though, supporting truly extensible record schemes. This |
|
788 admits operations that are polymorphic with respect to record |
|
789 extension, yielding ``object-oriented'' effects like (single) |
|
790 inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more |
|
791 details on object-oriented verification and record subtyping in HOL. |
|
792 *} |
|
793 |
|
794 |
|
795 subsection {* Basic concepts *} |
|
796 |
|
797 text {* |
|
798 Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records |
|
799 at the level of terms and types. The notation is as follows: |
|
800 |
|
801 \begin{center} |
|
802 \begin{tabular}{l|l|l} |
|
803 & record terms & record types \\ \hline |
|
804 fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\ |
|
805 schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} & |
|
806 @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\ |
|
807 \end{tabular} |
|
808 \end{center} |
|
809 |
|
810 \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text |
|
811 "(| x = a |)"}. |
|
812 |
|
813 A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value |
|
814 @{text a} and field @{text y} of value @{text b}. The corresponding |
|
815 type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"} |
|
816 and @{text "b :: B"}. |
|
817 |
|
818 A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields |
|
819 @{text x} and @{text y} as before, but also possibly further fields |
|
820 as indicated by the ``@{text "\<dots>"}'' notation (which is actually part |
|
821 of the syntax). The improper field ``@{text "\<dots>"}'' of a record |
|
822 scheme is called the \emph{more part}. Logically it is just a free |
|
823 variable, which is occasionally referred to as ``row variable'' in |
|
824 the literature. The more part of a record scheme may be |
|
825 instantiated by zero or more further components. For example, the |
|
826 previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z = |
|
827 c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part. |
|
828 Fixed records are special instances of record schemes, where |
|
829 ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"} |
|
830 element. In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation |
|
831 for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}. |
|
832 |
|
833 \medskip Two key observations make extensible records in a simply |
|
834 typed language like HOL work out: |
|
835 |
|
836 \begin{enumerate} |
|
837 |
|
838 \item the more part is internalized, as a free term or type |
|
839 variable, |
|
840 |
|
841 \item field names are externalized, they cannot be accessed within |
|
842 the logic as first-class values. |
|
843 |
|
844 \end{enumerate} |
|
845 |
|
846 \medskip In Isabelle/HOL record types have to be defined explicitly, |
|
847 fixing their field names and types, and their (optional) parent |
|
848 record. Afterwards, records may be formed using above syntax, while |
|
849 obeying the canonical order of fields as given by their declaration. |
|
850 The record package provides several standard operations like |
|
851 selectors and updates. The common setup for various generic proof |
|
852 tools enable succinct reasoning patterns. See also the Isabelle/HOL |
|
853 tutorial \cite{isabelle-hol-book} for further instructions on using |
|
854 records in practice. |
|
855 *} |
|
856 |
|
857 |
|
858 subsection {* Record specifications *} |
|
859 |
|
860 text {* |
|
861 \begin{matharray}{rcl} |
|
862 @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
863 \end{matharray} |
|
864 |
|
865 @{rail " |
|
866 @@{command (HOL) record} @{syntax typespec_sorts} '=' \\ |
|
867 (@{syntax type} '+')? (constdecl +) |
|
868 ; |
|
869 constdecl: @{syntax name} '::' @{syntax type} @{syntax mixfix}? |
|
870 "} |
|
871 |
|
872 \begin{description} |
|
873 |
|
874 \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1 |
|
875 \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"}, |
|
876 derived from the optional parent record @{text "\<tau>"} by adding new |
|
877 field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc. |
|
878 |
|
879 The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be |
|
880 covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>, |
|
881 \<alpha>\<^sub>m"}. Type constructor @{text t} has to be new, while @{text |
|
882 \<tau>} needs to specify an instance of an existing record type. At |
|
883 least one new field @{text "c\<^sub>i"} has to be specified. |
|
884 Basically, field names need to belong to a unique record. This is |
|
885 not a real restriction in practice, since fields are qualified by |
|
886 the record name internally. |
|
887 |
|
888 The parent record specification @{text \<tau>} is optional; if omitted |
|
889 @{text t} becomes a root record. The hierarchy of all records |
|
890 declared within a theory context forms a forest structure, i.e.\ a |
|
891 set of trees starting with a root record each. There is no way to |
|
892 merge multiple parent records! |
|
893 |
|
894 For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a |
|
895 type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 :: |
|
896 \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text |
|
897 "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for |
|
898 @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> :: |
|
899 \<zeta>\<rparr>"}. |
|
900 |
|
901 \end{description} |
|
902 *} |
|
903 |
|
904 |
|
905 subsection {* Record operations *} |
|
906 |
|
907 text {* |
|
908 Any record definition of the form presented above produces certain |
|
909 standard operations. Selectors and updates are provided for any |
|
910 field, including the improper one ``@{text more}''. There are also |
|
911 cumulative record constructor functions. To simplify the |
|
912 presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>, |
|
913 \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 :: |
|
914 \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}. |
|
915 |
|
916 \medskip \textbf{Selectors} and \textbf{updates} are available for |
|
917 any field (including ``@{text more}''): |
|
918 |
|
919 \begin{matharray}{lll} |
|
920 @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\ |
|
921 @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\ |
|
922 \end{matharray} |
|
923 |
|
924 There is special syntax for application of updates: @{text "r\<lparr>x := |
|
925 a\<rparr>"} abbreviates term @{text "x_update a r"}. Further notation for |
|
926 repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z := |
|
927 c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}. Note that |
|
928 because of postfix notation the order of fields shown here is |
|
929 reverse than in the actual term. Since repeated updates are just |
|
930 function applications, fields may be freely permuted in @{text "\<lparr>x |
|
931 := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned. |
|
932 Thus commutativity of independent updates can be proven within the |
|
933 logic for any two fields, but not as a general theorem. |
|
934 |
|
935 \medskip The \textbf{make} operation provides a cumulative record |
|
936 constructor function: |
|
937 |
|
938 \begin{matharray}{lll} |
|
939 @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ |
|
940 \end{matharray} |
|
941 |
|
942 \medskip We now reconsider the case of non-root records, which are |
|
943 derived of some parent. In general, the latter may depend on |
|
944 another parent as well, resulting in a list of \emph{ancestor |
|
945 records}. Appending the lists of fields of all ancestors results in |
|
946 a certain field prefix. The record package automatically takes care |
|
947 of this by lifting operations over this context of ancestor fields. |
|
948 Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor |
|
949 fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"}, |
|
950 the above record operations will get the following types: |
|
951 |
|
952 \medskip |
|
953 \begin{tabular}{lll} |
|
954 @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\ |
|
955 @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> |
|
956 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> |
|
957 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\ |
|
958 @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> |
|
959 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ |
|
960 \end{tabular} |
|
961 \medskip |
|
962 |
|
963 \noindent Some further operations address the extension aspect of a |
|
964 derived record scheme specifically: @{text "t.fields"} produces a |
|
965 record fragment consisting of exactly the new fields introduced here |
|
966 (the result may serve as a more part elsewhere); @{text "t.extend"} |
|
967 takes a fixed record and adds a given more part; @{text |
|
968 "t.truncate"} restricts a record scheme to a fixed record. |
|
969 |
|
970 \medskip |
|
971 \begin{tabular}{lll} |
|
972 @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ |
|
973 @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow> |
|
974 \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\ |
|
975 @{text "t.truncate"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\ |
|
976 \end{tabular} |
|
977 \medskip |
|
978 |
|
979 \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide |
|
980 for root records. |
|
981 *} |
|
982 |
|
983 |
|
984 subsection {* Derived rules and proof tools *} |
|
985 |
|
986 text {* |
|
987 The record package proves several results internally, declaring |
|
988 these facts to appropriate proof tools. This enables users to |
|
989 reason about record structures quite conveniently. Assume that |
|
990 @{text t} is a record type as specified above. |
|
991 |
|
992 \begin{enumerate} |
|
993 |
|
994 \item Standard conversions for selectors or updates applied to |
|
995 record constructor terms are made part of the default Simplifier |
|
996 context; thus proofs by reduction of basic operations merely require |
|
997 the @{method simp} method without further arguments. These rules |
|
998 are available as @{text "t.simps"}, too. |
|
999 |
|
1000 \item Selectors applied to updated records are automatically reduced |
|
1001 by an internal simplification procedure, which is also part of the |
|
1002 standard Simplifier setup. |
|
1003 |
|
1004 \item Inject equations of a form analogous to @{prop "(x, y) = (x', |
|
1005 y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical |
|
1006 Reasoner as @{attribute iff} rules. These rules are available as |
|
1007 @{text "t.iffs"}. |
|
1008 |
|
1009 \item The introduction rule for record equality analogous to @{text |
|
1010 "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier, |
|
1011 and as the basic rule context as ``@{attribute intro}@{text "?"}''. |
|
1012 The rule is called @{text "t.equality"}. |
|
1013 |
|
1014 \item Representations of arbitrary record expressions as canonical |
|
1015 constructor terms are provided both in @{method cases} and @{method |
|
1016 induct} format (cf.\ the generic proof methods of the same name, |
|
1017 \secref{sec:cases-induct}). Several variations are available, for |
|
1018 fixed records, record schemes, more parts etc. |
|
1019 |
|
1020 The generic proof methods are sufficiently smart to pick the most |
|
1021 sensible rule according to the type of the indicated record |
|
1022 expression: users just need to apply something like ``@{text "(cases |
|
1023 r)"}'' to a certain proof problem. |
|
1024 |
|
1025 \item The derived record operations @{text "t.make"}, @{text |
|
1026 "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not} |
|
1027 treated automatically, but usually need to be expanded by hand, |
|
1028 using the collective fact @{text "t.defs"}. |
|
1029 |
|
1030 \end{enumerate} |
|
1031 *} |
|
1032 |
|
1033 |
|
1034 subsubsection {* Examples *} |
|
1035 |
|
1036 text {* See @{file "~~/src/HOL/ex/Records.thy"}, for example. *} |
|
1037 |
|
1038 |
|
1039 section {* Adhoc tuples *} |
|
1040 |
|
1041 text {* |
|
1042 \begin{matharray}{rcl} |
|
1043 @{attribute_def (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\ |
|
1044 \end{matharray} |
|
1045 |
|
1046 @{rail " |
|
1047 @@{attribute (HOL) split_format} ('(' 'complete' ')')? |
|
1048 "} |
|
1049 |
|
1050 \begin{description} |
|
1051 |
|
1052 \item @{attribute (HOL) split_format}\ @{text "(complete)"} causes |
|
1053 arguments in function applications to be represented canonically |
|
1054 according to their tuple type structure. |
|
1055 |
|
1056 Note that this operation tends to invent funny names for new local |
|
1057 parameters introduced. |
|
1058 |
|
1059 \end{description} |
|
1060 *} |
|
1061 |
|
1062 |
|
1063 section {* Typedef axiomatization \label{sec:hol-typedef} *} |
|
1064 |
|
1065 text {* |
|
1066 \begin{matharray}{rcl} |
|
1067 @{command_def (HOL) "typedef"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\ |
|
1068 \end{matharray} |
|
1069 |
|
1070 A Gordon/HOL-style type definition is a certain axiom scheme that |
|
1071 identifies a new type with a subset of an existing type. More |
|
1072 precisely, the new type is defined by exhibiting an existing type |
|
1073 @{text \<tau>}, a set @{text "A :: \<tau> set"}, and a theorem that proves |
|
1074 @{prop "\<exists>x. x \<in> A"}. Thus @{text A} is a non-empty subset of @{text |
|
1075 \<tau>}, and the new type denotes this subset. New functions are |
|
1076 postulated that establish an isomorphism between the new type and |
|
1077 the subset. In general, the type @{text \<tau>} may involve type |
|
1078 variables @{text "\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n"} which means that the type definition |
|
1079 produces a type constructor @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} depending on |
|
1080 those type arguments. |
|
1081 |
|
1082 The axiomatization can be considered a ``definition'' in the sense |
|
1083 of the particular set-theoretic interpretation of HOL |
|
1084 \cite{pitts93}, where the universe of types is required to be |
|
1085 downwards-closed wrt.\ arbitrary non-empty subsets. Thus genuinely |
|
1086 new types introduced by @{command "typedef"} stay within the range |
|
1087 of HOL models by construction. Note that @{command_ref |
|
1088 type_synonym} from Isabelle/Pure merely introduces syntactic |
|
1089 abbreviations, without any logical significance. |
|
1090 |
|
1091 @{rail " |
|
1092 @@{command (HOL) typedef} alt_name? abs_type '=' rep_set |
|
1093 ; |
|
1094 |
|
1095 alt_name: '(' (@{syntax name} | @'open' | @'open' @{syntax name}) ')' |
|
1096 ; |
|
1097 abs_type: @{syntax typespec_sorts} @{syntax mixfix}? |
|
1098 ; |
|
1099 rep_set: @{syntax term} (@'morphisms' @{syntax name} @{syntax name})? |
|
1100 "} |
|
1101 |
|
1102 \begin{description} |
|
1103 |
|
1104 \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"} |
|
1105 axiomatizes a type definition in the background theory of the |
|
1106 current context, depending on a non-emptiness result of the set |
|
1107 @{text A} that needs to be proven here. The set @{text A} may |
|
1108 contain type variables @{text "\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n"} as specified on the LHS, |
|
1109 but no term variables. |
|
1110 |
|
1111 Even though a local theory specification, the newly introduced type |
|
1112 constructor cannot depend on parameters or assumptions of the |
|
1113 context: this is structurally impossible in HOL. In contrast, the |
|
1114 non-emptiness proof may use local assumptions in unusual situations, |
|
1115 which could result in different interpretations in target contexts: |
|
1116 the meaning of the bijection between the representing set @{text A} |
|
1117 and the new type @{text t} may then change in different application |
|
1118 contexts. |
|
1119 |
|
1120 By default, @{command (HOL) "typedef"} defines both a type |
|
1121 constructor @{text t} for the new type, and a term constant @{text |
|
1122 t} for the representing set within the old type. Use the ``@{text |
|
1123 "(open)"}'' option to suppress a separate constant definition |
|
1124 altogether. The injection from type to set is called @{text Rep_t}, |
|
1125 its inverse @{text Abs_t}, unless explicit @{keyword (HOL) |
|
1126 "morphisms"} specification provides alternative names. |
|
1127 |
|
1128 The core axiomatization uses the locale predicate @{const |
|
1129 type_definition} as defined in Isabelle/HOL. Various basic |
|
1130 consequences of that are instantiated accordingly, re-using the |
|
1131 locale facts with names derived from the new type constructor. Thus |
|
1132 the generic @{thm type_definition.Rep} is turned into the specific |
|
1133 @{text "Rep_t"}, for example. |
|
1134 |
|
1135 Theorems @{thm type_definition.Rep}, @{thm |
|
1136 type_definition.Rep_inverse}, and @{thm type_definition.Abs_inverse} |
|
1137 provide the most basic characterization as a corresponding |
|
1138 injection/surjection pair (in both directions). The derived rules |
|
1139 @{thm type_definition.Rep_inject} and @{thm |
|
1140 type_definition.Abs_inject} provide a more convenient version of |
|
1141 injectivity, suitable for automated proof tools (e.g.\ in |
|
1142 declarations involving @{attribute simp} or @{attribute iff}). |
|
1143 Furthermore, the rules @{thm type_definition.Rep_cases}~/ @{thm |
|
1144 type_definition.Rep_induct}, and @{thm type_definition.Abs_cases}~/ |
|
1145 @{thm type_definition.Abs_induct} provide alternative views on |
|
1146 surjectivity. These rules are already declared as set or type rules |
|
1147 for the generic @{method cases} and @{method induct} methods, |
|
1148 respectively. |
|
1149 |
|
1150 An alternative name for the set definition (and other derived |
|
1151 entities) may be specified in parentheses; the default is to use |
|
1152 @{text t} directly. |
|
1153 |
|
1154 \end{description} |
|
1155 |
|
1156 \begin{warn} |
|
1157 If you introduce a new type axiomatically, i.e.\ via @{command_ref |
|
1158 typedecl} and @{command_ref axiomatization}, the minimum requirement |
|
1159 is that it has a non-empty model, to avoid immediate collapse of the |
|
1160 HOL logic. Moreover, one needs to demonstrate that the |
|
1161 interpretation of such free-form axiomatizations can coexist with |
|
1162 that of the regular @{command_def typedef} scheme, and any extension |
|
1163 that other people might have introduced elsewhere (e.g.\ in HOLCF |
|
1164 \cite{MuellerNvOS99}). |
|
1165 \end{warn} |
|
1166 *} |
|
1167 |
|
1168 subsubsection {* Examples *} |
|
1169 |
|
1170 text {* Type definitions permit the introduction of abstract data |
|
1171 types in a safe way, namely by providing models based on already |
|
1172 existing types. Given some abstract axiomatic description @{text P} |
|
1173 of a type, this involves two steps: |
|
1174 |
|
1175 \begin{enumerate} |
|
1176 |
|
1177 \item Find an appropriate type @{text \<tau>} and subset @{text A} which |
|
1178 has the desired properties @{text P}, and make a type definition |
|
1179 based on this representation. |
|
1180 |
|
1181 \item Prove that @{text P} holds for @{text \<tau>} by lifting @{text P} |
|
1182 from the representation. |
|
1183 |
|
1184 \end{enumerate} |
|
1185 |
|
1186 You can later forget about the representation and work solely in |
|
1187 terms of the abstract properties @{text P}. |
|
1188 |
|
1189 \medskip The following trivial example pulls a three-element type |
|
1190 into existence within the formal logical environment of HOL. *} |
|
1191 |
|
1192 typedef three = "{(True, True), (True, False), (False, True)}" |
|
1193 by blast |
|
1194 |
|
1195 definition "One = Abs_three (True, True)" |
|
1196 definition "Two = Abs_three (True, False)" |
|
1197 definition "Three = Abs_three (False, True)" |
|
1198 |
|
1199 lemma three_distinct: "One \<noteq> Two" "One \<noteq> Three" "Two \<noteq> Three" |
|
1200 by (simp_all add: One_def Two_def Three_def Abs_three_inject three_def) |
|
1201 |
|
1202 lemma three_cases: |
|
1203 fixes x :: three obtains "x = One" | "x = Two" | "x = Three" |
|
1204 by (cases x) (auto simp: One_def Two_def Three_def Abs_three_inject three_def) |
|
1205 |
|
1206 text {* Note that such trivial constructions are better done with |
|
1207 derived specification mechanisms such as @{command datatype}: *} |
|
1208 |
|
1209 datatype three' = One' | Two' | Three' |
|
1210 |
|
1211 text {* This avoids re-doing basic definitions and proofs from the |
|
1212 primitive @{command typedef} above. *} |
|
1213 |
|
1214 |
|
1215 section {* Functorial structure of types *} |
|
1216 |
|
1217 text {* |
|
1218 \begin{matharray}{rcl} |
|
1219 @{command_def (HOL) "enriched_type"} & : & @{text "local_theory \<rightarrow> proof(prove)"} |
|
1220 \end{matharray} |
|
1221 |
|
1222 @{rail " |
|
1223 @@{command (HOL) enriched_type} (@{syntax name} ':')? @{syntax term} |
|
1224 ; |
|
1225 "} |
|
1226 |
|
1227 \begin{description} |
|
1228 |
|
1229 \item @{command (HOL) "enriched_type"}~@{text "prefix: m"} allows to |
|
1230 prove and register properties about the functorial structure of type |
|
1231 constructors. These properties then can be used by other packages |
|
1232 to deal with those type constructors in certain type constructions. |
|
1233 Characteristic theorems are noted in the current local theory. By |
|
1234 default, they are prefixed with the base name of the type |
|
1235 constructor, an explicit prefix can be given alternatively. |
|
1236 |
|
1237 The given term @{text "m"} is considered as \emph{mapper} for the |
|
1238 corresponding type constructor and must conform to the following |
|
1239 type pattern: |
|
1240 |
|
1241 \begin{matharray}{lll} |
|
1242 @{text "m"} & @{text "::"} & |
|
1243 @{text "\<sigma>\<^isub>1 \<Rightarrow> \<dots> \<sigma>\<^isub>k \<Rightarrow> (\<^vec>\<alpha>\<^isub>n) t \<Rightarrow> (\<^vec>\<beta>\<^isub>n) t"} \\ |
|
1244 \end{matharray} |
|
1245 |
|
1246 \noindent where @{text t} is the type constructor, @{text |
|
1247 "\<^vec>\<alpha>\<^isub>n"} and @{text "\<^vec>\<beta>\<^isub>n"} are distinct |
|
1248 type variables free in the local theory and @{text "\<sigma>\<^isub>1"}, |
|
1249 \ldots, @{text "\<sigma>\<^isub>k"} is a subsequence of @{text "\<alpha>\<^isub>1 \<Rightarrow> |
|
1250 \<beta>\<^isub>1"}, @{text "\<beta>\<^isub>1 \<Rightarrow> \<alpha>\<^isub>1"}, \ldots, |
|
1251 @{text "\<alpha>\<^isub>n \<Rightarrow> \<beta>\<^isub>n"}, @{text "\<beta>\<^isub>n \<Rightarrow> |
|
1252 \<alpha>\<^isub>n"}. |
|
1253 |
|
1254 \end{description} |
|
1255 *} |
|
1256 |
|
1257 |
|
1258 section {* Transfer package *} |
|
1259 |
|
1260 text {* |
|
1261 \begin{matharray}{rcl} |
|
1262 @{method_def (HOL) "transfer"} & : & @{text method} \\ |
|
1263 @{method_def (HOL) "transfer'"} & : & @{text method} \\ |
|
1264 @{method_def (HOL) "transfer_prover"} & : & @{text method} \\ |
|
1265 @{attribute_def (HOL) "transfer_rule"} & : & @{text attribute} \\ |
|
1266 @{attribute_def (HOL) "relator_eq"} & : & @{text attribute} \\ |
|
1267 \end{matharray} |
|
1268 |
|
1269 \begin{description} |
|
1270 |
|
1271 \item @{method (HOL) "transfer"} method replaces the current subgoal |
|
1272 with a logically equivalent one that uses different types and |
|
1273 constants. The replacement of types and constants is guided by the |
|
1274 database of transfer rules. Goals are generalized over all free |
|
1275 variables by default; this is necessary for variables whose types |
|
1276 change, but can be overridden for specific variables with e.g. |
|
1277 @{text "transfer fixing: x y z"}. |
|
1278 |
|
1279 \item @{method (HOL) "transfer'"} is a variant of @{method (HOL) |
|
1280 transfer} that allows replacing a subgoal with one that is |
|
1281 logically stronger (rather than equivalent). For example, a |
|
1282 subgoal involving equality on a quotient type could be replaced |
|
1283 with a subgoal involving equality (instead of the corresponding |
|
1284 equivalence relation) on the underlying raw type. |
|
1285 |
|
1286 \item @{method (HOL) "transfer_prover"} method assists with proving |
|
1287 a transfer rule for a new constant, provided the constant is |
|
1288 defined in terms of other constants that already have transfer |
|
1289 rules. It should be applied after unfolding the constant |
|
1290 definitions. |
|
1291 |
|
1292 \item @{attribute (HOL) "transfer_rule"} attribute maintains a |
|
1293 collection of transfer rules, which relate constants at two |
|
1294 different types. Typical transfer rules may relate different type |
|
1295 instances of the same polymorphic constant, or they may relate an |
|
1296 operation on a raw type to a corresponding operation on an |
|
1297 abstract type (quotient or subtype). For example: |
|
1298 |
|
1299 @{text "((A ===> B) ===> list_all2 A ===> list_all2 B) map map"}\\ |
|
1300 @{text "(cr_int ===> cr_int ===> cr_int) (\<lambda>(x,y) (u,v). (x+u, y+v)) plus"} |
|
1301 |
|
1302 Lemmas involving predicates on relations can also be registered |
|
1303 using the same attribute. For example: |
|
1304 |
|
1305 @{text "bi_unique A \<Longrightarrow> (list_all2 A ===> op =) distinct distinct"}\\ |
|
1306 @{text "\<lbrakk>bi_unique A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (prod_rel A B)"} |
|
1307 |
|
1308 \item @{attribute (HOL) relator_eq} attribute collects identity laws |
|
1309 for relators of various type constructors, e.g. @{text "list_all2 |
|
1310 (op =) = (op =)"}. The @{method (HOL) transfer} method uses these |
|
1311 lemmas to infer transfer rules for non-polymorphic constants on |
|
1312 the fly. |
|
1313 |
|
1314 \end{description} |
|
1315 |
|
1316 *} |
|
1317 |
|
1318 |
|
1319 section {* Lifting package *} |
|
1320 |
|
1321 text {* |
|
1322 \begin{matharray}{rcl} |
|
1323 @{command_def (HOL) "setup_lifting"} & : & @{text "local_theory \<rightarrow> local_theory"}\\ |
|
1324 @{command_def (HOL) "lift_definition"} & : & @{text "local_theory \<rightarrow> proof(prove)"}\\ |
|
1325 @{command_def (HOL) "print_quotmaps"} & : & @{text "context \<rightarrow>"}\\ |
|
1326 @{command_def (HOL) "print_quotients"} & : & @{text "context \<rightarrow>"}\\ |
|
1327 @{attribute_def (HOL) "quot_map"} & : & @{text attribute} \\ |
|
1328 @{attribute_def (HOL) "invariant_commute"} & : & @{text attribute} \\ |
|
1329 \end{matharray} |
|
1330 |
|
1331 @{rail " |
|
1332 @@{command (HOL) setup_lifting} ('(' 'no_abs_code' ')')? \\ |
|
1333 @{syntax thmref} @{syntax thmref}?; |
|
1334 "} |
|
1335 |
|
1336 @{rail " |
|
1337 @@{command (HOL) lift_definition} @{syntax name} '::' @{syntax type} @{syntax mixfix}? \\ |
|
1338 'is' @{syntax term}; |
|
1339 "} |
|
1340 |
|
1341 \begin{description} |
|
1342 |
|
1343 \item @{command (HOL) "setup_lifting"} Sets up the Lifting package |
|
1344 to work with a user-defined type. The user must provide either a |
|
1345 quotient theorem @{text "Quotient R Abs Rep T"} or a |
|
1346 type_definition theorem @{text "type_definition Rep Abs A"}. The |
|
1347 package configures transfer rules for equality and quantifiers on |
|
1348 the type, and sets up the @{command_def (HOL) "lift_definition"} |
|
1349 command to work with the type. In the case of a quotient theorem, |
|
1350 an optional theorem @{text "reflp R"} can be provided as a second |
|
1351 argument. This allows the package to generate stronger transfer |
|
1352 rules. |
|
1353 |
|
1354 @{command (HOL) "setup_lifting"} is called automatically if a |
|
1355 quotient type is defined by the command @{command (HOL) |
|
1356 "quotient_type"} from the Quotient package. |
|
1357 |
|
1358 If @{command (HOL) "setup_lifting"} is called with a |
|
1359 type_definition theorem, the abstract type implicitly defined by |
|
1360 the theorem is declared as an abstract type in the code |
|
1361 generator. This allows @{command (HOL) "lift_definition"} to |
|
1362 register (generated) code certificate theorems as abstract code |
|
1363 equations in the code generator. The option @{text "no_abs_code"} |
|
1364 of the command @{command (HOL) "setup_lifting"} can turn off that |
|
1365 behavior and causes that code certificate theorems generated by |
|
1366 @{command (HOL) "lift_definition"} are not registred as abstract |
|
1367 code equations. |
|
1368 |
|
1369 \item @{command (HOL) "lift_definition"} @{text "f :: \<tau> is t"} |
|
1370 Defines a new function @{text f} with an abstract type @{text \<tau>} |
|
1371 in terms of a corresponding operation @{text t} on a |
|
1372 representation type. The term @{text t} doesn't have to be |
|
1373 necessarily a constant but it can be any term. |
|
1374 |
|
1375 Users must discharge a respectfulness proof obligation when each |
|
1376 constant is defined. For a type copy, i.e. a typedef with @{text |
|
1377 UNIV}, the proof is discharged automatically. The obligation is |
|
1378 presented in a user-friendly, readable form. A respectfulness |
|
1379 theorem in the standard format @{text f.rsp} and a transfer rule |
|
1380 @{text f.tranfer} for the Transfer package are generated by the |
|
1381 package. |
|
1382 |
|
1383 Integration with code_abstype: For typedefs (e.g. subtypes |
|
1384 corresponding to a datatype invariant, such as dlist), @{command |
|
1385 (HOL) "lift_definition"} generates a code certificate theorem |
|
1386 @{text f.rep_eq} and sets up code generation for each constant. |
|
1387 |
|
1388 \item @{command (HOL) "print_quotmaps"} prints stored quotient map |
|
1389 theorems. |
|
1390 |
|
1391 \item @{command (HOL) "print_quotients"} prints stored quotient |
|
1392 theorems. |
|
1393 |
|
1394 \item @{attribute (HOL) quot_map} registers a quotient map |
|
1395 theorem. For examples see @{file |
|
1396 "~~/src/HOL/Library/Quotient_List.thy"} or other Quotient_*.thy |
|
1397 files. |
|
1398 |
|
1399 \item @{attribute (HOL) invariant_commute} registers a theorem which |
|
1400 shows a relationship between the constant @{text |
|
1401 Lifting.invariant} (used for internal encoding of proper subtypes) |
|
1402 and a relator. Such theorems allows the package to hide @{text |
|
1403 Lifting.invariant} from a user in a user-readable form of a |
|
1404 respectfulness theorem. For examples see @{file |
|
1405 "~~/src/HOL/Library/Quotient_List.thy"} or other Quotient_*.thy |
|
1406 files. |
|
1407 |
|
1408 \end{description} |
|
1409 *} |
|
1410 |
|
1411 |
|
1412 section {* Quotient types *} |
|
1413 |
|
1414 text {* |
|
1415 \begin{matharray}{rcl} |
|
1416 @{command_def (HOL) "quotient_type"} & : & @{text "local_theory \<rightarrow> proof(prove)"}\\ |
|
1417 @{command_def (HOL) "quotient_definition"} & : & @{text "local_theory \<rightarrow> proof(prove)"}\\ |
|
1418 @{command_def (HOL) "print_quotmapsQ3"} & : & @{text "context \<rightarrow>"}\\ |
|
1419 @{command_def (HOL) "print_quotientsQ3"} & : & @{text "context \<rightarrow>"}\\ |
|
1420 @{command_def (HOL) "print_quotconsts"} & : & @{text "context \<rightarrow>"}\\ |
|
1421 @{method_def (HOL) "lifting"} & : & @{text method} \\ |
|
1422 @{method_def (HOL) "lifting_setup"} & : & @{text method} \\ |
|
1423 @{method_def (HOL) "descending"} & : & @{text method} \\ |
|
1424 @{method_def (HOL) "descending_setup"} & : & @{text method} \\ |
|
1425 @{method_def (HOL) "partiality_descending"} & : & @{text method} \\ |
|
1426 @{method_def (HOL) "partiality_descending_setup"} & : & @{text method} \\ |
|
1427 @{method_def (HOL) "regularize"} & : & @{text method} \\ |
|
1428 @{method_def (HOL) "injection"} & : & @{text method} \\ |
|
1429 @{method_def (HOL) "cleaning"} & : & @{text method} \\ |
|
1430 @{attribute_def (HOL) "quot_thm"} & : & @{text attribute} \\ |
|
1431 @{attribute_def (HOL) "quot_lifted"} & : & @{text attribute} \\ |
|
1432 @{attribute_def (HOL) "quot_respect"} & : & @{text attribute} \\ |
|
1433 @{attribute_def (HOL) "quot_preserve"} & : & @{text attribute} \\ |
|
1434 \end{matharray} |
|
1435 |
|
1436 The quotient package defines a new quotient type given a raw type |
|
1437 and a partial equivalence relation. It also includes automation for |
|
1438 transporting definitions and theorems. It can automatically produce |
|
1439 definitions and theorems on the quotient type, given the |
|
1440 corresponding constants and facts on the raw type. |
|
1441 |
|
1442 @{rail " |
|
1443 @@{command (HOL) quotient_type} (spec + @'and'); |
|
1444 |
|
1445 spec: @{syntax typespec} @{syntax mixfix}? '=' \\ |
|
1446 @{syntax type} '/' ('partial' ':')? @{syntax term} \\ |
|
1447 (@'morphisms' @{syntax name} @{syntax name})?; |
|
1448 "} |
|
1449 |
|
1450 @{rail " |
|
1451 @@{command (HOL) quotient_definition} constdecl? @{syntax thmdecl}? \\ |
|
1452 @{syntax term} 'is' @{syntax term}; |
|
1453 |
|
1454 constdecl: @{syntax name} ('::' @{syntax type})? @{syntax mixfix}? |
|
1455 "} |
|
1456 |
|
1457 @{rail " |
|
1458 @@{method (HOL) lifting} @{syntax thmrefs}? |
|
1459 ; |
|
1460 |
|
1461 @@{method (HOL) lifting_setup} @{syntax thmrefs}? |
|
1462 ; |
|
1463 "} |
|
1464 |
|
1465 \begin{description} |
|
1466 |
|
1467 \item @{command (HOL) "quotient_type"} defines quotient types. The |
|
1468 injection from a quotient type to a raw type is called @{text |
|
1469 rep_t}, its inverse @{text abs_t} unless explicit @{keyword (HOL) |
|
1470 "morphisms"} specification provides alternative names. @{command |
|
1471 (HOL) "quotient_type"} requires the user to prove that the relation |
|
1472 is an equivalence relation (predicate @{text equivp}), unless the |
|
1473 user specifies explicitely @{text partial} in which case the |
|
1474 obligation is @{text part_equivp}. A quotient defined with @{text |
|
1475 partial} is weaker in the sense that less things can be proved |
|
1476 automatically. |
|
1477 |
|
1478 \item @{command (HOL) "quotient_definition"} defines a constant on |
|
1479 the quotient type. |
|
1480 |
|
1481 \item @{command (HOL) "print_quotmapsQ3"} prints quotient map |
|
1482 functions. |
|
1483 |
|
1484 \item @{command (HOL) "print_quotientsQ3"} prints quotients. |
|
1485 |
|
1486 \item @{command (HOL) "print_quotconsts"} prints quotient constants. |
|
1487 |
|
1488 \item @{method (HOL) "lifting"} and @{method (HOL) "lifting_setup"} |
|
1489 methods match the current goal with the given raw theorem to be |
|
1490 lifted producing three new subgoals: regularization, injection and |
|
1491 cleaning subgoals. @{method (HOL) "lifting"} tries to apply the |
|
1492 heuristics for automatically solving these three subgoals and |
|
1493 leaves only the subgoals unsolved by the heuristics to the user as |
|
1494 opposed to @{method (HOL) "lifting_setup"} which leaves the three |
|
1495 subgoals unsolved. |
|
1496 |
|
1497 \item @{method (HOL) "descending"} and @{method (HOL) |
|
1498 "descending_setup"} try to guess a raw statement that would lift |
|
1499 to the current subgoal. Such statement is assumed as a new subgoal |
|
1500 and @{method (HOL) "descending"} continues in the same way as |
|
1501 @{method (HOL) "lifting"} does. @{method (HOL) "descending"} tries |
|
1502 to solve the arising regularization, injection and cleaning |
|
1503 subgoals with the analoguous method @{method (HOL) |
|
1504 "descending_setup"} which leaves the four unsolved subgoals. |
|
1505 |
|
1506 \item @{method (HOL) "partiality_descending"} finds the regularized |
|
1507 theorem that would lift to the current subgoal, lifts it and |
|
1508 leaves as a subgoal. This method can be used with partial |
|
1509 equivalence quotients where the non regularized statements would |
|
1510 not be true. @{method (HOL) "partiality_descending_setup"} leaves |
|
1511 the injection and cleaning subgoals unchanged. |
|
1512 |
|
1513 \item @{method (HOL) "regularize"} applies the regularization |
|
1514 heuristics to the current subgoal. |
|
1515 |
|
1516 \item @{method (HOL) "injection"} applies the injection heuristics |
|
1517 to the current goal using the stored quotient respectfulness |
|
1518 theorems. |
|
1519 |
|
1520 \item @{method (HOL) "cleaning"} applies the injection cleaning |
|
1521 heuristics to the current subgoal using the stored quotient |
|
1522 preservation theorems. |
|
1523 |
|
1524 \item @{attribute (HOL) quot_lifted} attribute tries to |
|
1525 automatically transport the theorem to the quotient type. |
|
1526 The attribute uses all the defined quotients types and quotient |
|
1527 constants often producing undesired results or theorems that |
|
1528 cannot be lifted. |
|
1529 |
|
1530 \item @{attribute (HOL) quot_respect} and @{attribute (HOL) |
|
1531 quot_preserve} attributes declare a theorem as a respectfulness |
|
1532 and preservation theorem respectively. These are stored in the |
|
1533 local theory store and used by the @{method (HOL) "injection"} |
|
1534 and @{method (HOL) "cleaning"} methods respectively. |
|
1535 |
|
1536 \item @{attribute (HOL) quot_thm} declares that a certain theorem |
|
1537 is a quotient extension theorem. Quotient extension theorems |
|
1538 allow for quotienting inside container types. Given a polymorphic |
|
1539 type that serves as a container, a map function defined for this |
|
1540 container using @{command (HOL) "enriched_type"} and a relation |
|
1541 map defined for for the container type, the quotient extension |
|
1542 theorem should be @{term "Quotient3 R Abs Rep \<Longrightarrow> Quotient3 |
|
1543 (rel_map R) (map Abs) (map Rep)"}. Quotient extension theorems |
|
1544 are stored in a database and are used all the steps of lifting |
|
1545 theorems. |
|
1546 |
|
1547 \end{description} |
|
1548 *} |
|
1549 |
|
1550 |
|
1551 section {* Coercive subtyping *} |
|
1552 |
|
1553 text {* |
|
1554 \begin{matharray}{rcl} |
|
1555 @{attribute_def (HOL) coercion} & : & @{text attribute} \\ |
|
1556 @{attribute_def (HOL) coercion_enabled} & : & @{text attribute} \\ |
|
1557 @{attribute_def (HOL) coercion_map} & : & @{text attribute} \\ |
|
1558 \end{matharray} |
|
1559 |
|
1560 Coercive subtyping allows the user to omit explicit type |
|
1561 conversions, also called \emph{coercions}. Type inference will add |
|
1562 them as necessary when parsing a term. See |
|
1563 \cite{traytel-berghofer-nipkow-2011} for details. |
|
1564 |
|
1565 @{rail " |
|
1566 @@{attribute (HOL) coercion} (@{syntax term})? |
|
1567 ; |
|
1568 "} |
|
1569 @{rail " |
|
1570 @@{attribute (HOL) coercion_map} (@{syntax term})? |
|
1571 ; |
|
1572 "} |
|
1573 |
|
1574 \begin{description} |
|
1575 |
|
1576 \item @{attribute (HOL) "coercion"}~@{text "f"} registers a new |
|
1577 coercion function @{text "f :: \<sigma>\<^isub>1 \<Rightarrow> \<sigma>\<^isub>2"} where @{text "\<sigma>\<^isub>1"} and |
|
1578 @{text "\<sigma>\<^isub>2"} are type constructors without arguments. Coercions are |
|
1579 composed by the inference algorithm if needed. Note that the type |
|
1580 inference algorithm is complete only if the registered coercions |
|
1581 form a lattice. |
|
1582 |
|
1583 \item @{attribute (HOL) "coercion_map"}~@{text "map"} registers a |
|
1584 new map function to lift coercions through type constructors. The |
|
1585 function @{text "map"} must conform to the following type pattern |
|
1586 |
|
1587 \begin{matharray}{lll} |
|
1588 @{text "map"} & @{text "::"} & |
|
1589 @{text "f\<^isub>1 \<Rightarrow> \<dots> \<Rightarrow> f\<^isub>n \<Rightarrow> (\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>n) t \<Rightarrow> (\<beta>\<^isub>1, \<dots>, \<beta>\<^isub>n) t"} \\ |
|
1590 \end{matharray} |
|
1591 |
|
1592 where @{text "t"} is a type constructor and @{text "f\<^isub>i"} is of type |
|
1593 @{text "\<alpha>\<^isub>i \<Rightarrow> \<beta>\<^isub>i"} or @{text "\<beta>\<^isub>i \<Rightarrow> \<alpha>\<^isub>i"}. Registering a map function |
|
1594 overwrites any existing map function for this particular type |
|
1595 constructor. |
|
1596 |
|
1597 \item @{attribute (HOL) "coercion_enabled"} enables the coercion |
|
1598 inference algorithm. |
|
1599 |
|
1600 \end{description} |
|
1601 *} |
|
1602 |
|
1603 |
|
1604 section {* Arithmetic proof support *} |
|
1605 |
|
1606 text {* |
|
1607 \begin{matharray}{rcl} |
|
1608 @{method_def (HOL) arith} & : & @{text method} \\ |
|
1609 @{attribute_def (HOL) arith} & : & @{text attribute} \\ |
|
1610 @{attribute_def (HOL) arith_split} & : & @{text attribute} \\ |
|
1611 \end{matharray} |
|
1612 |
|
1613 \begin{description} |
|
1614 |
|
1615 \item @{method (HOL) arith} decides linear arithmetic problems (on |
|
1616 types @{text nat}, @{text int}, @{text real}). Any current facts |
|
1617 are inserted into the goal before running the procedure. |
|
1618 |
|
1619 \item @{attribute (HOL) arith} declares facts that are supplied to |
|
1620 the arithmetic provers implicitly. |
|
1621 |
|
1622 \item @{attribute (HOL) arith_split} attribute declares case split |
|
1623 rules to be expanded before @{method (HOL) arith} is invoked. |
|
1624 |
|
1625 \end{description} |
|
1626 |
|
1627 Note that a simpler (but faster) arithmetic prover is already |
|
1628 invoked by the Simplifier. |
|
1629 *} |
|
1630 |
|
1631 |
|
1632 section {* Intuitionistic proof search *} |
|
1633 |
|
1634 text {* |
|
1635 \begin{matharray}{rcl} |
|
1636 @{method_def (HOL) iprover} & : & @{text method} \\ |
|
1637 \end{matharray} |
|
1638 |
|
1639 @{rail " |
|
1640 @@{method (HOL) iprover} ( @{syntax rulemod} * ) |
|
1641 "} |
|
1642 |
|
1643 \begin{description} |
|
1644 |
|
1645 \item @{method (HOL) iprover} performs intuitionistic proof search, |
|
1646 depending on specifically declared rules from the context, or given |
|
1647 as explicit arguments. Chained facts are inserted into the goal |
|
1648 before commencing proof search. |
|
1649 |
|
1650 Rules need to be classified as @{attribute (Pure) intro}, |
|
1651 @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the |
|
1652 ``@{text "!"}'' indicator refers to ``safe'' rules, which may be |
|
1653 applied aggressively (without considering back-tracking later). |
|
1654 Rules declared with ``@{text "?"}'' are ignored in proof search (the |
|
1655 single-step @{method (Pure) rule} method still observes these). An |
|
1656 explicit weight annotation may be given as well; otherwise the |
|
1657 number of rule premises will be taken into account here. |
|
1658 |
|
1659 \end{description} |
|
1660 *} |
|
1661 |
|
1662 |
|
1663 section {* Model Elimination and Resolution *} |
|
1664 |
|
1665 text {* |
|
1666 \begin{matharray}{rcl} |
|
1667 @{method_def (HOL) "meson"} & : & @{text method} \\ |
|
1668 @{method_def (HOL) "metis"} & : & @{text method} \\ |
|
1669 \end{matharray} |
|
1670 |
|
1671 @{rail " |
|
1672 @@{method (HOL) meson} @{syntax thmrefs}? |
|
1673 ; |
|
1674 |
|
1675 @@{method (HOL) metis} |
|
1676 ('(' ('partial_types' | 'full_types' | 'no_types' | @{syntax name}) ')')? |
|
1677 @{syntax thmrefs}? |
|
1678 "} |
|
1679 |
|
1680 \begin{description} |
|
1681 |
|
1682 \item @{method (HOL) meson} implements Loveland's model elimination |
|
1683 procedure \cite{loveland-78}. See @{file |
|
1684 "~~/src/HOL/ex/Meson_Test.thy"} for examples. |
|
1685 |
|
1686 \item @{method (HOL) metis} combines ordered resolution and ordered |
|
1687 paramodulation to find first-order (or mildly higher-order) proofs. |
|
1688 The first optional argument specifies a type encoding; see the |
|
1689 Sledgehammer manual \cite{isabelle-sledgehammer} for details. The |
|
1690 directory @{file "~~/src/HOL/Metis_Examples"} contains several small |
|
1691 theories developed to a large extent using @{method (HOL) metis}. |
|
1692 |
|
1693 \end{description} |
|
1694 *} |
|
1695 |
|
1696 |
|
1697 section {* Coherent Logic *} |
|
1698 |
|
1699 text {* |
|
1700 \begin{matharray}{rcl} |
|
1701 @{method_def (HOL) "coherent"} & : & @{text method} \\ |
|
1702 \end{matharray} |
|
1703 |
|
1704 @{rail " |
|
1705 @@{method (HOL) coherent} @{syntax thmrefs}? |
|
1706 "} |
|
1707 |
|
1708 \begin{description} |
|
1709 |
|
1710 \item @{method (HOL) coherent} solves problems of \emph{Coherent |
|
1711 Logic} \cite{Bezem-Coquand:2005}, which covers applications in |
|
1712 confluence theory, lattice theory and projective geometry. See |
|
1713 @{file "~~/src/HOL/ex/Coherent.thy"} for some examples. |
|
1714 |
|
1715 \end{description} |
|
1716 *} |
|
1717 |
|
1718 |
|
1719 section {* Proving propositions *} |
|
1720 |
|
1721 text {* |
|
1722 In addition to the standard proof methods, a number of diagnosis |
|
1723 tools search for proofs and provide an Isar proof snippet on success. |
|
1724 These tools are available via the following commands. |
|
1725 |
|
1726 \begin{matharray}{rcl} |
|
1727 @{command_def (HOL) "solve_direct"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
|
1728 @{command_def (HOL) "try"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
|
1729 @{command_def (HOL) "try0"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
|
1730 @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
|
1731 @{command_def (HOL) "sledgehammer_params"} & : & @{text "theory \<rightarrow> theory"} |
|
1732 \end{matharray} |
|
1733 |
|
1734 @{rail " |
|
1735 @@{command (HOL) try} |
|
1736 ; |
|
1737 |
|
1738 @@{command (HOL) try0} ( ( ( 'simp' | 'intro' | 'elim' | 'dest' ) ':' @{syntax thmrefs} ) + ) ? |
|
1739 @{syntax nat}? |
|
1740 ; |
|
1741 |
|
1742 @@{command (HOL) sledgehammer} ( '[' args ']' )? facts? @{syntax nat}? |
|
1743 ; |
|
1744 |
|
1745 @@{command (HOL) sledgehammer_params} ( ( '[' args ']' ) ? ) |
|
1746 ; |
|
1747 |
|
1748 args: ( @{syntax name} '=' value + ',' ) |
|
1749 ; |
|
1750 |
|
1751 facts: '(' ( ( ( ( 'add' | 'del' ) ':' ) ? @{syntax thmrefs} ) + ) ? ')' |
|
1752 ; |
|
1753 "} % FIXME check args "value" |
|
1754 |
|
1755 \begin{description} |
|
1756 |
|
1757 \item @{command (HOL) "solve_direct"} checks whether the current |
|
1758 subgoals can be solved directly by an existing theorem. Duplicate |
|
1759 lemmas can be detected in this way. |
|
1760 |
|
1761 \item @{command (HOL) "try0"} attempts to prove a subgoal |
|
1762 using a combination of standard proof methods (@{method auto}, |
|
1763 @{method simp}, @{method blast}, etc.). Additional facts supplied |
|
1764 via @{text "simp:"}, @{text "intro:"}, @{text "elim:"}, and @{text |
|
1765 "dest:"} are passed to the appropriate proof methods. |
|
1766 |
|
1767 \item @{command (HOL) "try"} attempts to prove or disprove a subgoal |
|
1768 using a combination of provers and disprovers (@{command (HOL) |
|
1769 "solve_direct"}, @{command (HOL) "quickcheck"}, @{command (HOL) |
|
1770 "try0"}, @{command (HOL) "sledgehammer"}, @{command (HOL) |
|
1771 "nitpick"}). |
|
1772 |
|
1773 \item @{command (HOL) "sledgehammer"} attempts to prove a subgoal |
|
1774 using external automatic provers (resolution provers and SMT |
|
1775 solvers). See the Sledgehammer manual \cite{isabelle-sledgehammer} |
|
1776 for details. |
|
1777 |
|
1778 \item @{command (HOL) "sledgehammer_params"} changes @{command (HOL) |
|
1779 "sledgehammer"} configuration options persistently. |
|
1780 |
|
1781 \end{description} |
|
1782 *} |
|
1783 |
|
1784 |
|
1785 section {* Checking and refuting propositions *} |
|
1786 |
|
1787 text {* |
|
1788 Identifying incorrect propositions usually involves evaluation of |
|
1789 particular assignments and systematic counterexample search. This |
|
1790 is supported by the following commands. |
|
1791 |
|
1792 \begin{matharray}{rcl} |
|
1793 @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
|
1794 @{command_def (HOL) "values"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
|
1795 @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
|
1796 @{command_def (HOL) "refute"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
|
1797 @{command_def (HOL) "nitpick"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\ |
|
1798 @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
1799 @{command_def (HOL) "refute_params"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
1800 @{command_def (HOL) "nitpick_params"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
1801 @{command_def (HOL) "quickcheck_generator"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
1802 @{command_def (HOL) "find_unused_assms"} & : & @{text "context \<rightarrow>"} |
|
1803 \end{matharray} |
|
1804 |
|
1805 @{rail " |
|
1806 @@{command (HOL) value} ( '[' @{syntax name} ']' )? modes? @{syntax term} |
|
1807 ; |
|
1808 |
|
1809 @@{command (HOL) values} modes? @{syntax nat}? @{syntax term} |
|
1810 ; |
|
1811 |
|
1812 (@@{command (HOL) quickcheck} | @@{command (HOL) refute} | @@{command (HOL) nitpick}) |
|
1813 ( '[' args ']' )? @{syntax nat}? |
|
1814 ; |
|
1815 |
|
1816 (@@{command (HOL) quickcheck_params} | @@{command (HOL) refute_params} | |
|
1817 @@{command (HOL) nitpick_params}) ( '[' args ']' )? |
|
1818 ; |
|
1819 |
|
1820 @@{command (HOL) quickcheck_generator} @{syntax nameref} \\ |
|
1821 'operations:' ( @{syntax term} +) |
|
1822 ; |
|
1823 |
|
1824 @@{command (HOL) find_unused_assms} @{syntax name}? |
|
1825 ; |
|
1826 |
|
1827 modes: '(' (@{syntax name} +) ')' |
|
1828 ; |
|
1829 |
|
1830 args: ( @{syntax name} '=' value + ',' ) |
|
1831 ; |
|
1832 "} % FIXME check "value" |
|
1833 |
|
1834 \begin{description} |
|
1835 |
|
1836 \item @{command (HOL) "value"}~@{text t} evaluates and prints a |
|
1837 term; optionally @{text modes} can be specified, which are appended |
|
1838 to the current print mode; see \secref{sec:print-modes}. |
|
1839 Internally, the evaluation is performed by registered evaluators, |
|
1840 which are invoked sequentially until a result is returned. |
|
1841 Alternatively a specific evaluator can be selected using square |
|
1842 brackets; typical evaluators use the current set of code equations |
|
1843 to normalize and include @{text simp} for fully symbolic evaluation |
|
1844 using the simplifier, @{text nbe} for \emph{normalization by |
|
1845 evaluation} and \emph{code} for code generation in SML. |
|
1846 |
|
1847 \item @{command (HOL) "values"}~@{text t} enumerates a set |
|
1848 comprehension by evaluation and prints its values up to the given |
|
1849 number of solutions; optionally @{text modes} can be specified, |
|
1850 which are appended to the current print mode; see |
|
1851 \secref{sec:print-modes}. |
|
1852 |
|
1853 \item @{command (HOL) "quickcheck"} tests the current goal for |
|
1854 counterexamples using a series of assignments for its free |
|
1855 variables; by default the first subgoal is tested, an other can be |
|
1856 selected explicitly using an optional goal index. Assignments can |
|
1857 be chosen exhausting the search space upto a given size, or using a |
|
1858 fixed number of random assignments in the search space, or exploring |
|
1859 the search space symbolically using narrowing. By default, |
|
1860 quickcheck uses exhaustive testing. A number of configuration |
|
1861 options are supported for @{command (HOL) "quickcheck"}, notably: |
|
1862 |
|
1863 \begin{description} |
|
1864 |
|
1865 \item[@{text tester}] specifies which testing approach to apply. |
|
1866 There are three testers, @{text exhaustive}, @{text random}, and |
|
1867 @{text narrowing}. An unknown configuration option is treated as |
|
1868 an argument to tester, making @{text "tester ="} optional. When |
|
1869 multiple testers are given, these are applied in parallel. If no |
|
1870 tester is specified, quickcheck uses the testers that are set |
|
1871 active, i.e., configurations @{attribute |
|
1872 quickcheck_exhaustive_active}, @{attribute |
|
1873 quickcheck_random_active}, @{attribute |
|
1874 quickcheck_narrowing_active} are set to true. |
|
1875 |
|
1876 \item[@{text size}] specifies the maximum size of the search space |
|
1877 for assignment values. |
|
1878 |
|
1879 \item[@{text genuine_only}] sets quickcheck only to return genuine |
|
1880 counterexample, but not potentially spurious counterexamples due |
|
1881 to underspecified functions. |
|
1882 |
|
1883 \item[@{text abort_potential}] sets quickcheck to abort once it |
|
1884 found a potentially spurious counterexample and to not continue |
|
1885 to search for a further genuine counterexample. |
|
1886 For this option to be effective, the @{text genuine_only} option |
|
1887 must be set to false. |
|
1888 |
|
1889 \item[@{text eval}] takes a term or a list of terms and evaluates |
|
1890 these terms under the variable assignment found by quickcheck. |
|
1891 This option is currently only supported by the default |
|
1892 (exhaustive) tester. |
|
1893 |
|
1894 \item[@{text iterations}] sets how many sets of assignments are |
|
1895 generated for each particular size. |
|
1896 |
|
1897 \item[@{text no_assms}] specifies whether assumptions in |
|
1898 structured proofs should be ignored. |
|
1899 |
|
1900 \item[@{text locale}] specifies how to process conjectures in |
|
1901 a locale context, i.e., they can be interpreted or expanded. |
|
1902 The option is a whitespace-separated list of the two words |
|
1903 @{text interpret} and @{text expand}. The list determines the |
|
1904 order they are employed. The default setting is to first use |
|
1905 interpretations and then test the expanded conjecture. |
|
1906 The option is only provided as attribute declaration, but not |
|
1907 as parameter to the command. |
|
1908 |
|
1909 \item[@{text timeout}] sets the time limit in seconds. |
|
1910 |
|
1911 \item[@{text default_type}] sets the type(s) generally used to |
|
1912 instantiate type variables. |
|
1913 |
|
1914 \item[@{text report}] if set quickcheck reports how many tests |
|
1915 fulfilled the preconditions. |
|
1916 |
|
1917 \item[@{text use_subtype}] if set quickcheck automatically lifts |
|
1918 conjectures to registered subtypes if possible, and tests the |
|
1919 lifted conjecture. |
|
1920 |
|
1921 \item[@{text quiet}] if set quickcheck does not output anything |
|
1922 while testing. |
|
1923 |
|
1924 \item[@{text verbose}] if set quickcheck informs about the current |
|
1925 size and cardinality while testing. |
|
1926 |
|
1927 \item[@{text expect}] can be used to check if the user's |
|
1928 expectation was met (@{text no_expectation}, @{text |
|
1929 no_counterexample}, or @{text counterexample}). |
|
1930 |
|
1931 \end{description} |
|
1932 |
|
1933 These option can be given within square brackets. |
|
1934 |
|
1935 \item @{command (HOL) "quickcheck_params"} changes @{command (HOL) |
|
1936 "quickcheck"} configuration options persistently. |
|
1937 |
|
1938 \item @{command (HOL) "quickcheck_generator"} creates random and |
|
1939 exhaustive value generators for a given type and operations. It |
|
1940 generates values by using the operations as if they were |
|
1941 constructors of that type. |
|
1942 |
|
1943 \item @{command (HOL) "refute"} tests the current goal for |
|
1944 counterexamples using a reduction to SAT. The following |
|
1945 configuration options are supported: |
|
1946 |
|
1947 \begin{description} |
|
1948 |
|
1949 \item[@{text minsize}] specifies the minimum size (cardinality) of |
|
1950 the models to search for. |
|
1951 |
|
1952 \item[@{text maxsize}] specifies the maximum size (cardinality) of |
|
1953 the models to search for. Nonpositive values mean @{text "\<infinity>"}. |
|
1954 |
|
1955 \item[@{text maxvars}] specifies the maximum number of Boolean |
|
1956 variables to use when transforming the term into a propositional |
|
1957 formula. Nonpositive values mean @{text "\<infinity>"}. |
|
1958 |
|
1959 \item[@{text satsolver}] specifies the SAT solver to use. |
|
1960 |
|
1961 \item[@{text no_assms}] specifies whether assumptions in |
|
1962 structured proofs should be ignored. |
|
1963 |
|
1964 \item[@{text maxtime}] sets the time limit in seconds. |
|
1965 |
|
1966 \item[@{text expect}] can be used to check if the user's |
|
1967 expectation was met (@{text genuine}, @{text potential}, @{text |
|
1968 none}, or @{text unknown}). |
|
1969 |
|
1970 \end{description} |
|
1971 |
|
1972 These option can be given within square brackets. |
|
1973 |
|
1974 \item @{command (HOL) "refute_params"} changes @{command (HOL) |
|
1975 "refute"} configuration options persistently. |
|
1976 |
|
1977 \item @{command (HOL) "nitpick"} tests the current goal for |
|
1978 counterexamples using a reduction to first-order relational |
|
1979 logic. See the Nitpick manual \cite{isabelle-nitpick} for details. |
|
1980 |
|
1981 \item @{command (HOL) "nitpick_params"} changes @{command (HOL) |
|
1982 "nitpick"} configuration options persistently. |
|
1983 |
|
1984 \item @{command (HOL) "find_unused_assms"} finds potentially superfluous |
|
1985 assumptions in theorems using quickcheck. |
|
1986 It takes the theory name to be checked for superfluous assumptions as |
|
1987 optional argument. If not provided, it checks the current theory. |
|
1988 Options to the internal quickcheck invocations can be changed with |
|
1989 common configuration declarations. |
|
1990 |
|
1991 \end{description} |
|
1992 *} |
|
1993 |
|
1994 |
|
1995 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *} |
|
1996 |
|
1997 text {* |
|
1998 The following tools of Isabelle/HOL support cases analysis and |
|
1999 induction in unstructured tactic scripts; see also |
|
2000 \secref{sec:cases-induct} for proper Isar versions of similar ideas. |
|
2001 |
|
2002 \begin{matharray}{rcl} |
|
2003 @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
|
2004 @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\ |
|
2005 @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\ |
|
2006 @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
|
2007 \end{matharray} |
|
2008 |
|
2009 @{rail " |
|
2010 @@{method (HOL) case_tac} @{syntax goal_spec}? @{syntax term} rule? |
|
2011 ; |
|
2012 @@{method (HOL) induct_tac} @{syntax goal_spec}? (@{syntax insts} * @'and') rule? |
|
2013 ; |
|
2014 @@{method (HOL) ind_cases} (@{syntax prop}+) (@'for' (@{syntax name}+))? |
|
2015 ; |
|
2016 @@{command (HOL) inductive_cases} (@{syntax thmdecl}? (@{syntax prop}+) + @'and') |
|
2017 ; |
|
2018 |
|
2019 rule: 'rule' ':' @{syntax thmref} |
|
2020 "} |
|
2021 |
|
2022 \begin{description} |
|
2023 |
|
2024 \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit |
|
2025 to reason about inductive types. Rules are selected according to |
|
2026 the declarations by the @{attribute cases} and @{attribute induct} |
|
2027 attributes, cf.\ \secref{sec:cases-induct}. The @{command (HOL) |
|
2028 datatype} package already takes care of this. |
|
2029 |
|
2030 These unstructured tactics feature both goal addressing and dynamic |
|
2031 instantiation. Note that named rule cases are \emph{not} provided |
|
2032 as would be by the proper @{method cases} and @{method induct} proof |
|
2033 methods (see \secref{sec:cases-induct}). Unlike the @{method |
|
2034 induct} method, @{method induct_tac} does not handle structured rule |
|
2035 statements, only the compact object-logic conclusion of the subgoal |
|
2036 being addressed. |
|
2037 |
|
2038 \item @{method (HOL) ind_cases} and @{command (HOL) |
|
2039 "inductive_cases"} provide an interface to the internal @{ML_text |
|
2040 mk_cases} operation. Rules are simplified in an unrestricted |
|
2041 forward manner. |
|
2042 |
|
2043 While @{method (HOL) ind_cases} is a proof method to apply the |
|
2044 result immediately as elimination rules, @{command (HOL) |
|
2045 "inductive_cases"} provides case split theorems at the theory level |
|
2046 for later use. The @{keyword "for"} argument of the @{method (HOL) |
|
2047 ind_cases} method allows to specify a list of variables that should |
|
2048 be generalized before applying the resulting rule. |
|
2049 |
|
2050 \end{description} |
|
2051 *} |
|
2052 |
|
2053 |
|
2054 section {* Executable code *} |
|
2055 |
|
2056 text {* For validation purposes, it is often useful to \emph{execute} |
|
2057 specifications. In principle, execution could be simulated by |
|
2058 Isabelle's inference kernel, i.e. by a combination of resolution and |
|
2059 simplification. Unfortunately, this approach is rather inefficient. |
|
2060 A more efficient way of executing specifications is to translate |
|
2061 them into a functional programming language such as ML. |
|
2062 |
|
2063 Isabelle provides a generic framework to support code generation |
|
2064 from executable specifications. Isabelle/HOL instantiates these |
|
2065 mechanisms in a way that is amenable to end-user applications. Code |
|
2066 can be generated for functional programs (including overloading |
|
2067 using type classes) targeting SML \cite{SML}, OCaml \cite{OCaml}, |
|
2068 Haskell \cite{haskell-revised-report} and Scala |
|
2069 \cite{scala-overview-tech-report}. Conceptually, code generation is |
|
2070 split up in three steps: \emph{selection} of code theorems, |
|
2071 \emph{translation} into an abstract executable view and |
|
2072 \emph{serialization} to a specific \emph{target language}. |
|
2073 Inductive specifications can be executed using the predicate |
|
2074 compiler which operates within HOL. See \cite{isabelle-codegen} for |
|
2075 an introduction. |
|
2076 |
|
2077 \begin{matharray}{rcl} |
|
2078 @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
|
2079 @{attribute_def (HOL) code} & : & @{text attribute} \\ |
|
2080 @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2081 @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2082 @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
|
2083 @{attribute_def (HOL) code_unfold} & : & @{text attribute} \\ |
|
2084 @{attribute_def (HOL) code_post} & : & @{text attribute} \\ |
|
2085 @{attribute_def (HOL) code_abbrev} & : & @{text attribute} \\ |
|
2086 @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
|
2087 @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
|
2088 @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ |
|
2089 @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2090 @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2091 @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2092 @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2093 @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2094 @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2095 @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2096 @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2097 @{command_def (HOL) "code_reflect"} & : & @{text "theory \<rightarrow> theory"} \\ |
|
2098 @{command_def (HOL) "code_pred"} & : & @{text "theory \<rightarrow> proof(prove)"} |
|
2099 \end{matharray} |
|
2100 |
|
2101 @{rail " |
|
2102 @@{command (HOL) export_code} ( constexpr + ) \\ |
|
2103 ( ( @'in' target ( @'module_name' @{syntax string} ) ? \\ |
|
2104 ( @'file' ( @{syntax string} | '-' ) ) ? ( '(' args ')' ) ?) + ) ? |
|
2105 ; |
|
2106 |
|
2107 const: @{syntax term} |
|
2108 ; |
|
2109 |
|
2110 constexpr: ( const | 'name._' | '_' ) |
|
2111 ; |
|
2112 |
|
2113 typeconstructor: @{syntax nameref} |
|
2114 ; |
|
2115 |
|
2116 class: @{syntax nameref} |
|
2117 ; |
|
2118 |
|
2119 target: 'SML' | 'OCaml' | 'Haskell' | 'Scala' |
|
2120 ; |
|
2121 |
|
2122 @@{attribute (HOL) code} ( 'del' | 'abstype' | 'abstract' )? |
|
2123 ; |
|
2124 |
|
2125 @@{command (HOL) code_abort} ( const + ) |
|
2126 ; |
|
2127 |
|
2128 @@{command (HOL) code_datatype} ( const + ) |
|
2129 ; |
|
2130 |
|
2131 @@{attribute (HOL) code_unfold} ( 'del' ) ? |
|
2132 ; |
|
2133 |
|
2134 @@{attribute (HOL) code_post} ( 'del' ) ? |
|
2135 ; |
|
2136 |
|
2137 @@{attribute (HOL) code_abbrev} |
|
2138 ; |
|
2139 |
|
2140 @@{command (HOL) code_thms} ( constexpr + ) ? |
|
2141 ; |
|
2142 |
|
2143 @@{command (HOL) code_deps} ( constexpr + ) ? |
|
2144 ; |
|
2145 |
|
2146 @@{command (HOL) code_const} (const + @'and') \\ |
|
2147 ( ( '(' target ( syntax ? + @'and' ) ')' ) + ) |
|
2148 ; |
|
2149 |
|
2150 @@{command (HOL) code_type} (typeconstructor + @'and') \\ |
|
2151 ( ( '(' target ( syntax ? + @'and' ) ')' ) + ) |
|
2152 ; |
|
2153 |
|
2154 @@{command (HOL) code_class} (class + @'and') \\ |
|
2155 ( ( '(' target \\ ( @{syntax string} ? + @'and' ) ')' ) + ) |
|
2156 ; |
|
2157 |
|
2158 @@{command (HOL) code_instance} (( typeconstructor '::' class ) + @'and') \\ |
|
2159 ( ( '(' target ( '-' ? + @'and' ) ')' ) + ) |
|
2160 ; |
|
2161 |
|
2162 @@{command (HOL) code_reserved} target ( @{syntax string} + ) |
|
2163 ; |
|
2164 |
|
2165 @@{command (HOL) code_monad} const const target |
|
2166 ; |
|
2167 |
|
2168 @@{command (HOL) code_include} target ( @{syntax string} ( @{syntax string} | '-') ) |
|
2169 ; |
|
2170 |
|
2171 @@{command (HOL) code_modulename} target ( ( @{syntax string} @{syntax string} ) + ) |
|
2172 ; |
|
2173 |
|
2174 @@{command (HOL) code_reflect} @{syntax string} \\ |
|
2175 ( @'datatypes' ( @{syntax string} '=' ( '_' | ( @{syntax string} + '|' ) + @'and' ) ) ) ? \\ |
|
2176 ( @'functions' ( @{syntax string} + ) ) ? ( @'file' @{syntax string} ) ? |
|
2177 ; |
|
2178 |
|
2179 @@{command (HOL) code_pred} \\( '(' @'modes' ':' modedecl ')')? \\ const |
|
2180 ; |
|
2181 |
|
2182 syntax: @{syntax string} | ( @'infix' | @'infixl' | @'infixr' ) @{syntax nat} @{syntax string} |
|
2183 ; |
|
2184 |
|
2185 modedecl: (modes | ((const ':' modes) \\ |
|
2186 (@'and' ((const ':' modes @'and') +))?)) |
|
2187 ; |
|
2188 |
|
2189 modes: mode @'as' const |
|
2190 "} |
|
2191 |
|
2192 \begin{description} |
|
2193 |
|
2194 \item @{command (HOL) "export_code"} generates code for a given list |
|
2195 of constants in the specified target language(s). If no |
|
2196 serialization instruction is given, only abstract code is generated |
|
2197 internally. |
|
2198 |
|
2199 Constants may be specified by giving them literally, referring to |
|
2200 all executable contants within a certain theory by giving @{text |
|
2201 "name.*"}, or referring to \emph{all} executable constants currently |
|
2202 available by giving @{text "*"}. |
|
2203 |
|
2204 By default, for each involved theory one corresponding name space |
|
2205 module is generated. Alternativly, a module name may be specified |
|
2206 after the @{keyword "module_name"} keyword; then \emph{all} code is |
|
2207 placed in this module. |
|
2208 |
|
2209 For \emph{SML}, \emph{OCaml} and \emph{Scala} the file specification |
|
2210 refers to a single file; for \emph{Haskell}, it refers to a whole |
|
2211 directory, where code is generated in multiple files reflecting the |
|
2212 module hierarchy. Omitting the file specification denotes standard |
|
2213 output. |
|
2214 |
|
2215 Serializers take an optional list of arguments in parentheses. For |
|
2216 \emph{SML} and \emph{OCaml}, ``@{text no_signatures}`` omits |
|
2217 explicit module signatures. |
|
2218 |
|
2219 For \emph{Haskell} a module name prefix may be given using the |
|
2220 ``@{text "root:"}'' argument; ``@{text string_classes}'' adds a |
|
2221 ``@{verbatim "deriving (Read, Show)"}'' clause to each appropriate |
|
2222 datatype declaration. |
|
2223 |
|
2224 \item @{attribute (HOL) code} explicitly selects (or with option |
|
2225 ``@{text "del"}'' deselects) a code equation for code generation. |
|
2226 Usually packages introducing code equations provide a reasonable |
|
2227 default setup for selection. Variants @{text "code abstype"} and |
|
2228 @{text "code abstract"} declare abstract datatype certificates or |
|
2229 code equations on abstract datatype representations respectively. |
|
2230 |
|
2231 \item @{command (HOL) "code_abort"} declares constants which are not |
|
2232 required to have a definition by means of code equations; if needed |
|
2233 these are implemented by program abort instead. |
|
2234 |
|
2235 \item @{command (HOL) "code_datatype"} specifies a constructor set |
|
2236 for a logical type. |
|
2237 |
|
2238 \item @{command (HOL) "print_codesetup"} gives an overview on |
|
2239 selected code equations and code generator datatypes. |
|
2240 |
|
2241 \item @{attribute (HOL) code_unfold} declares (or with option |
|
2242 ``@{text "del"}'' removes) theorems which during preprocessing |
|
2243 are applied as rewrite rules to any code equation or evaluation |
|
2244 input. |
|
2245 |
|
2246 \item @{attribute (HOL) code_post} declares (or with option ``@{text |
|
2247 "del"}'' removes) theorems which are applied as rewrite rules to any |
|
2248 result of an evaluation. |
|
2249 |
|
2250 \item @{attribute (HOL) code_abbrev} declares equations which are |
|
2251 applied as rewrite rules to any result of an evaluation and |
|
2252 symmetrically during preprocessing to any code equation or evaluation |
|
2253 input. |
|
2254 |
|
2255 \item @{command (HOL) "print_codeproc"} prints the setup of the code |
|
2256 generator preprocessor. |
|
2257 |
|
2258 \item @{command (HOL) "code_thms"} prints a list of theorems |
|
2259 representing the corresponding program containing all given |
|
2260 constants after preprocessing. |
|
2261 |
|
2262 \item @{command (HOL) "code_deps"} visualizes dependencies of |
|
2263 theorems representing the corresponding program containing all given |
|
2264 constants after preprocessing. |
|
2265 |
|
2266 \item @{command (HOL) "code_const"} associates a list of constants |
|
2267 with target-specific serializations; omitting a serialization |
|
2268 deletes an existing serialization. |
|
2269 |
|
2270 \item @{command (HOL) "code_type"} associates a list of type |
|
2271 constructors with target-specific serializations; omitting a |
|
2272 serialization deletes an existing serialization. |
|
2273 |
|
2274 \item @{command (HOL) "code_class"} associates a list of classes |
|
2275 with target-specific class names; omitting a serialization deletes |
|
2276 an existing serialization. This applies only to \emph{Haskell}. |
|
2277 |
|
2278 \item @{command (HOL) "code_instance"} declares a list of type |
|
2279 constructor / class instance relations as ``already present'' for a |
|
2280 given target. Omitting a ``@{text "-"}'' deletes an existing |
|
2281 ``already present'' declaration. This applies only to |
|
2282 \emph{Haskell}. |
|
2283 |
|
2284 \item @{command (HOL) "code_reserved"} declares a list of names as |
|
2285 reserved for a given target, preventing it to be shadowed by any |
|
2286 generated code. |
|
2287 |
|
2288 \item @{command (HOL) "code_monad"} provides an auxiliary mechanism |
|
2289 to generate monadic code for Haskell. |
|
2290 |
|
2291 \item @{command (HOL) "code_include"} adds arbitrary named content |
|
2292 (``include'') to generated code. A ``@{text "-"}'' as last argument |
|
2293 will remove an already added ``include''. |
|
2294 |
|
2295 \item @{command (HOL) "code_modulename"} declares aliasings from one |
|
2296 module name onto another. |
|
2297 |
|
2298 \item @{command (HOL) "code_reflect"} without a ``@{text "file"}'' |
|
2299 argument compiles code into the system runtime environment and |
|
2300 modifies the code generator setup that future invocations of system |
|
2301 runtime code generation referring to one of the ``@{text |
|
2302 "datatypes"}'' or ``@{text "functions"}'' entities use these |
|
2303 precompiled entities. With a ``@{text "file"}'' argument, the |
|
2304 corresponding code is generated into that specified file without |
|
2305 modifying the code generator setup. |
|
2306 |
|
2307 \item @{command (HOL) "code_pred"} creates code equations for a |
|
2308 predicate given a set of introduction rules. Optional mode |
|
2309 annotations determine which arguments are supposed to be input or |
|
2310 output. If alternative introduction rules are declared, one must |
|
2311 prove a corresponding elimination rule. |
|
2312 |
|
2313 \end{description} |
|
2314 *} |
|
2315 |
|
2316 |
|
2317 section {* Definition by specification \label{sec:hol-specification} *} |
|
2318 |
|
2319 text {* |
|
2320 \begin{matharray}{rcl} |
|
2321 @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ |
|
2322 @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\ |
|
2323 \end{matharray} |
|
2324 |
|
2325 @{rail " |
|
2326 (@@{command (HOL) specification} | @@{command (HOL) ax_specification}) |
|
2327 '(' (decl +) ')' \\ (@{syntax thmdecl}? @{syntax prop} +) |
|
2328 ; |
|
2329 decl: ((@{syntax name} ':')? @{syntax term} '(' @'overloaded' ')'?) |
|
2330 "} |
|
2331 |
|
2332 \begin{description} |
|
2333 |
|
2334 \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a |
|
2335 goal stating the existence of terms with the properties specified to |
|
2336 hold for the constants given in @{text decls}. After finishing the |
|
2337 proof, the theory will be augmented with definitions for the given |
|
2338 constants, as well as with theorems stating the properties for these |
|
2339 constants. |
|
2340 |
|
2341 \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up |
|
2342 a goal stating the existence of terms with the properties specified |
|
2343 to hold for the constants given in @{text decls}. After finishing |
|
2344 the proof, the theory will be augmented with axioms expressing the |
|
2345 properties given in the first place. |
|
2346 |
|
2347 \item @{text decl} declares a constant to be defined by the |
|
2348 specification given. The definition for the constant @{text c} is |
|
2349 bound to the name @{text c_def} unless a theorem name is given in |
|
2350 the declaration. Overloaded constants should be declared as such. |
|
2351 |
|
2352 \end{description} |
|
2353 |
|
2354 Whether to use @{command (HOL) "specification"} or @{command (HOL) |
|
2355 "ax_specification"} is to some extent a matter of style. @{command |
|
2356 (HOL) "specification"} introduces no new axioms, and so by |
|
2357 construction cannot introduce inconsistencies, whereas @{command |
|
2358 (HOL) "ax_specification"} does introduce axioms, but only after the |
|
2359 user has explicitly proven it to be safe. A practical issue must be |
|
2360 considered, though: After introducing two constants with the same |
|
2361 properties using @{command (HOL) "specification"}, one can prove |
|
2362 that the two constants are, in fact, equal. If this might be a |
|
2363 problem, one should use @{command (HOL) "ax_specification"}. |
|
2364 *} |
|
2365 |
|
2366 end |
|