1 % |
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2 \begin{isabellebody}% |
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3 \def\isabellecontext{proof}% |
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4 % |
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5 \isadelimtheory |
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6 \isanewline |
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7 \isanewline |
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8 \isanewline |
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9 % |
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10 \endisadelimtheory |
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11 % |
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12 \isatagtheory |
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13 \isacommand{theory}\isamarkupfalse% |
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14 \ {\isachardoublequoteopen}proof{\isachardoublequoteclose}\ \isakeyword{imports}\ base\ \isakeyword{begin}% |
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15 \endisatagtheory |
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16 {\isafoldtheory}% |
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17 % |
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18 \isadelimtheory |
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19 % |
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20 \endisadelimtheory |
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21 % |
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22 \isamarkupchapter{Structured proofs% |
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23 } |
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24 \isamarkuptrue% |
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25 % |
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26 \isamarkupsection{Variables \label{sec:variables}% |
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27 } |
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28 \isamarkuptrue% |
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29 % |
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30 \begin{isamarkuptext}% |
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31 Any variable that is not explicitly bound by \isa{{\isasymlambda}}-abstraction |
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32 is considered as ``free''. Logically, free variables act like |
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33 outermost universal quantification at the sequent level: \isa{A\isactrlisub {\isadigit{1}}{\isacharparenleft}x{\isacharparenright}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ A\isactrlisub n{\isacharparenleft}x{\isacharparenright}\ {\isasymturnstile}\ B{\isacharparenleft}x{\isacharparenright}} means that the result |
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34 holds \emph{for all} values of \isa{x}. Free variables for |
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35 terms (not types) can be fully internalized into the logic: \isa{{\isasymturnstile}\ B{\isacharparenleft}x{\isacharparenright}} and \isa{{\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ B{\isacharparenleft}x{\isacharparenright}} are interchangeable, provided |
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36 that \isa{x} does not occur elsewhere in the context. |
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37 Inspecting \isa{{\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ B{\isacharparenleft}x{\isacharparenright}} more closely, we see that inside the |
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38 quantifier, \isa{x} is essentially ``arbitrary, but fixed'', |
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39 while from outside it appears as a place-holder for instantiation |
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40 (thanks to \isa{{\isasymAnd}} elimination). |
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41 |
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42 The Pure logic represents the idea of variables being either inside |
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43 or outside the current scope by providing separate syntactic |
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44 categories for \emph{fixed variables} (e.g.\ \isa{x}) vs.\ |
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45 \emph{schematic variables} (e.g.\ \isa{{\isacharquery}x}). Incidently, a |
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46 universal result \isa{{\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ B{\isacharparenleft}x{\isacharparenright}} has the HHF normal form \isa{{\isasymturnstile}\ B{\isacharparenleft}{\isacharquery}x{\isacharparenright}}, which represents its generality nicely without requiring |
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47 an explicit quantifier. The same principle works for type |
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48 variables: \isa{{\isasymturnstile}\ B{\isacharparenleft}{\isacharquery}{\isasymalpha}{\isacharparenright}} represents the idea of ``\isa{{\isasymturnstile}\ {\isasymforall}{\isasymalpha}{\isachardot}\ B{\isacharparenleft}{\isasymalpha}{\isacharparenright}}'' without demanding a truly polymorphic framework. |
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49 |
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50 \medskip Additional care is required to treat type variables in a |
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51 way that facilitates type-inference. In principle, term variables |
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52 depend on type variables, which means that type variables would have |
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53 to be declared first. For example, a raw type-theoretic framework |
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54 would demand the context to be constructed in stages as follows: |
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55 \isa{{\isasymGamma}\ {\isacharequal}\ {\isasymalpha}{\isacharcolon}\ type{\isacharcomma}\ x{\isacharcolon}\ {\isasymalpha}{\isacharcomma}\ a{\isacharcolon}\ A{\isacharparenleft}x\isactrlisub {\isasymalpha}{\isacharparenright}}. |
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56 |
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57 We allow a slightly less formalistic mode of operation: term |
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58 variables \isa{x} are fixed without specifying a type yet |
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59 (essentially \emph{all} potential occurrences of some instance |
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60 \isa{x\isactrlisub {\isasymtau}} are fixed); the first occurrence of \isa{x} |
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61 within a specific term assigns its most general type, which is then |
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62 maintained consistently in the context. The above example becomes |
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63 \isa{{\isasymGamma}\ {\isacharequal}\ x{\isacharcolon}\ term{\isacharcomma}\ {\isasymalpha}{\isacharcolon}\ type{\isacharcomma}\ A{\isacharparenleft}x\isactrlisub {\isasymalpha}{\isacharparenright}}, where type \isa{{\isasymalpha}} is fixed \emph{after} term \isa{x}, and the constraint |
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64 \isa{x\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}} is an implicit consequence of the occurrence of |
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65 \isa{x\isactrlisub {\isasymalpha}} in the subsequent proposition. |
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66 |
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67 This twist of dependencies is also accommodated by the reverse |
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68 operation of exporting results from a context: a type variable |
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69 \isa{{\isasymalpha}} is considered fixed as long as it occurs in some fixed |
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70 term variable of the context. For example, exporting \isa{x{\isacharcolon}\ term{\isacharcomma}\ {\isasymalpha}{\isacharcolon}\ type\ {\isasymturnstile}\ x\isactrlisub {\isasymalpha}\ {\isacharequal}\ x\isactrlisub {\isasymalpha}} produces in the first step |
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71 \isa{x{\isacharcolon}\ term\ {\isasymturnstile}\ x\isactrlisub {\isasymalpha}\ {\isacharequal}\ x\isactrlisub {\isasymalpha}} for fixed \isa{{\isasymalpha}}, |
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72 and only in the second step \isa{{\isasymturnstile}\ {\isacharquery}x\isactrlisub {\isacharquery}\isactrlisub {\isasymalpha}\ {\isacharequal}\ {\isacharquery}x\isactrlisub {\isacharquery}\isactrlisub {\isasymalpha}} for schematic \isa{{\isacharquery}x} and \isa{{\isacharquery}{\isasymalpha}}. |
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73 |
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74 \medskip The Isabelle/Isar proof context manages the gory details of |
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75 term vs.\ type variables, with high-level principles for moving the |
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76 frontier between fixed and schematic variables. |
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77 |
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78 The \isa{add{\isacharunderscore}fixes} operation explictly declares fixed |
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79 variables; the \isa{declare{\isacharunderscore}term} operation absorbs a term into |
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80 a context by fixing new type variables and adding syntactic |
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81 constraints. |
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82 |
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83 The \isa{export} operation is able to perform the main work of |
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84 generalizing term and type variables as sketched above, assuming |
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85 that fixing variables and terms have been declared properly. |
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86 |
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87 There \isa{import} operation makes a generalized fact a genuine |
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88 part of the context, by inventing fixed variables for the schematic |
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89 ones. The effect can be reversed by using \isa{export} later, |
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90 potentially with an extended context; the result is equivalent to |
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91 the original modulo renaming of schematic variables. |
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92 |
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93 The \isa{focus} operation provides a variant of \isa{import} |
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94 for nested propositions (with explicit quantification): \isa{{\isasymAnd}x\isactrlisub {\isadigit{1}}\ {\isasymdots}\ x\isactrlisub n{\isachardot}\ B{\isacharparenleft}x\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlisub n{\isacharparenright}} is |
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95 decomposed by inventing fixed variables \isa{x\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlisub n} for the body.% |
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96 \end{isamarkuptext}% |
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97 \isamarkuptrue% |
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98 % |
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99 \isadelimmlref |
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100 % |
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101 \endisadelimmlref |
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102 % |
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103 \isatagmlref |
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104 % |
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105 \begin{isamarkuptext}% |
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106 \begin{mldecls} |
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107 \indexml{Variable.add\_fixes}\verb|Variable.add_fixes: |\isasep\isanewline% |
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108 \verb| string list -> Proof.context -> string list * Proof.context| \\ |
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109 \indexml{Variable.variant\_fixes}\verb|Variable.variant_fixes: |\isasep\isanewline% |
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110 \verb| string list -> Proof.context -> string list * Proof.context| \\ |
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111 \indexml{Variable.declare\_term}\verb|Variable.declare_term: term -> Proof.context -> Proof.context| \\ |
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112 \indexml{Variable.declare\_constraints}\verb|Variable.declare_constraints: term -> Proof.context -> Proof.context| \\ |
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113 \indexml{Variable.export}\verb|Variable.export: Proof.context -> Proof.context -> thm list -> thm list| \\ |
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114 \indexml{Variable.polymorphic}\verb|Variable.polymorphic: Proof.context -> term list -> term list| \\ |
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115 \indexml{Variable.import\_thms}\verb|Variable.import_thms: bool -> thm list -> Proof.context ->|\isasep\isanewline% |
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116 \verb| ((ctyp list * cterm list) * thm list) * Proof.context| \\ |
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117 \indexml{Variable.focus}\verb|Variable.focus: cterm -> Proof.context -> (cterm list * cterm) * Proof.context| \\ |
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118 \end{mldecls} |
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119 |
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120 \begin{description} |
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121 |
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122 \item \verb|Variable.add_fixes|~\isa{xs\ ctxt} fixes term |
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123 variables \isa{xs}, returning the resulting internal names. By |
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124 default, the internal representation coincides with the external |
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125 one, which also means that the given variables must not be fixed |
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126 already. There is a different policy within a local proof body: the |
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127 given names are just hints for newly invented Skolem variables. |
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128 |
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129 \item \verb|Variable.variant_fixes| is similar to \verb|Variable.add_fixes|, but always produces fresh variants of the given |
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130 names. |
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131 |
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132 \item \verb|Variable.declare_term|~\isa{t\ ctxt} declares term |
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133 \isa{t} to belong to the context. This automatically fixes new |
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134 type variables, but not term variables. Syntactic constraints for |
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135 type and term variables are declared uniformly, though. |
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136 |
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137 \item \verb|Variable.declare_constraints|~\isa{t\ ctxt} declares |
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138 syntactic constraints from term \isa{t}, without making it part |
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139 of the context yet. |
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140 |
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141 \item \verb|Variable.export|~\isa{inner\ outer\ thms} generalizes |
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142 fixed type and term variables in \isa{thms} according to the |
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143 difference of the \isa{inner} and \isa{outer} context, |
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144 following the principles sketched above. |
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145 |
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146 \item \verb|Variable.polymorphic|~\isa{ctxt\ ts} generalizes type |
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147 variables in \isa{ts} as far as possible, even those occurring |
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148 in fixed term variables. The default policy of type-inference is to |
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149 fix newly introduced type variables, which is essentially reversed |
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150 with \verb|Variable.polymorphic|: here the given terms are detached |
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151 from the context as far as possible. |
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152 |
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153 \item \verb|Variable.import_thms|~\isa{open\ thms\ ctxt} invents fixed |
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154 type and term variables for the schematic ones occurring in \isa{thms}. The \isa{open} flag indicates whether the fixed names |
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155 should be accessible to the user, otherwise newly introduced names |
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156 are marked as ``internal'' (\secref{sec:names}). |
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157 |
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158 \item \verb|Variable.focus|~\isa{B} decomposes the outermost \isa{{\isasymAnd}} prefix of proposition \isa{B}. |
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159 |
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160 \end{description}% |
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161 \end{isamarkuptext}% |
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162 \isamarkuptrue% |
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163 % |
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164 \endisatagmlref |
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165 {\isafoldmlref}% |
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166 % |
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167 \isadelimmlref |
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168 % |
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169 \endisadelimmlref |
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170 % |
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171 \isamarkupsection{Assumptions \label{sec:assumptions}% |
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172 } |
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173 \isamarkuptrue% |
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174 % |
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175 \begin{isamarkuptext}% |
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176 An \emph{assumption} is a proposition that it is postulated in the |
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177 current context. Local conclusions may use assumptions as |
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178 additional facts, but this imposes implicit hypotheses that weaken |
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179 the overall statement. |
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180 |
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181 Assumptions are restricted to fixed non-schematic statements, i.e.\ |
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182 all generality needs to be expressed by explicit quantifiers. |
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183 Nevertheless, the result will be in HHF normal form with outermost |
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184 quantifiers stripped. For example, by assuming \isa{{\isasymAnd}x\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}{\isachardot}\ P\ x} we get \isa{{\isasymAnd}x\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}{\isachardot}\ P\ x\ {\isasymturnstile}\ P\ {\isacharquery}x} for schematic \isa{{\isacharquery}x} |
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185 of fixed type \isa{{\isasymalpha}}. Local derivations accumulate more and |
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186 more explicit references to hypotheses: \isa{A\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ A\isactrlisub n\ {\isasymturnstile}\ B} where \isa{A\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ A\isactrlisub n} needs to |
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187 be covered by the assumptions of the current context. |
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188 |
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189 \medskip The \isa{add{\isacharunderscore}assms} operation augments the context by |
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190 local assumptions, which are parameterized by an arbitrary \isa{export} rule (see below). |
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191 |
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192 The \isa{export} operation moves facts from a (larger) inner |
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193 context into a (smaller) outer context, by discharging the |
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194 difference of the assumptions as specified by the associated export |
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195 rules. Note that the discharged portion is determined by the |
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196 difference contexts, not the facts being exported! There is a |
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197 separate flag to indicate a goal context, where the result is meant |
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198 to refine an enclosing sub-goal of a structured proof state (cf.\ |
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199 \secref{sec:isar-proof-state}). |
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200 |
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201 \medskip The most basic export rule discharges assumptions directly |
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202 by means of the \isa{{\isasymLongrightarrow}} introduction rule: |
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203 \[ |
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204 \infer[(\isa{{\isasymLongrightarrow}{\isacharunderscore}intro})]{\isa{{\isasymGamma}\ {\isacharbackslash}\ A\ {\isasymturnstile}\ A\ {\isasymLongrightarrow}\ B}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B}} |
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205 \] |
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206 |
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207 The variant for goal refinements marks the newly introduced |
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208 premises, which causes the canonical Isar goal refinement scheme to |
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209 enforce unification with local premises within the goal: |
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210 \[ |
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211 \infer[(\isa{{\isacharhash}{\isasymLongrightarrow}{\isacharunderscore}intro})]{\isa{{\isasymGamma}\ {\isacharbackslash}\ A\ {\isasymturnstile}\ {\isacharhash}A\ {\isasymLongrightarrow}\ B}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B}} |
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212 \] |
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213 |
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214 \medskip Alternative versions of assumptions may perform arbitrary |
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215 transformations on export, as long as the corresponding portion of |
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216 hypotheses is removed from the given facts. For example, a local |
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217 definition works by fixing \isa{x} and assuming \isa{x\ {\isasymequiv}\ t}, |
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218 with the following export rule to reverse the effect: |
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219 \[ |
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220 \infer[(\isa{{\isasymequiv}{\isacharminus}expand})]{\isa{{\isasymGamma}\ {\isacharbackslash}\ x\ {\isasymequiv}\ t\ {\isasymturnstile}\ B\ t}}{\isa{{\isasymGamma}\ {\isasymturnstile}\ B\ x}} |
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221 \] |
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222 This works, because the assumption \isa{x\ {\isasymequiv}\ t} was introduced in |
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223 a context with \isa{x} being fresh, so \isa{x} does not |
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224 occur in \isa{{\isasymGamma}} here.% |
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225 \end{isamarkuptext}% |
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226 \isamarkuptrue% |
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227 % |
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228 \isadelimmlref |
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229 % |
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230 \endisadelimmlref |
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231 % |
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232 \isatagmlref |
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233 % |
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234 \begin{isamarkuptext}% |
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235 \begin{mldecls} |
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236 \indexmltype{Assumption.export}\verb|type Assumption.export| \\ |
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237 \indexml{Assumption.assume}\verb|Assumption.assume: cterm -> thm| \\ |
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238 \indexml{Assumption.add\_assms}\verb|Assumption.add_assms: Assumption.export ->|\isasep\isanewline% |
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239 \verb| cterm list -> Proof.context -> thm list * Proof.context| \\ |
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240 \indexml{Assumption.add\_assumes}\verb|Assumption.add_assumes: |\isasep\isanewline% |
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241 \verb| cterm list -> Proof.context -> thm list * Proof.context| \\ |
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242 \indexml{Assumption.export}\verb|Assumption.export: bool -> Proof.context -> Proof.context -> thm -> thm| \\ |
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243 \end{mldecls} |
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244 |
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245 \begin{description} |
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246 |
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247 \item \verb|Assumption.export| represents arbitrary export |
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248 rules, which is any function of type \verb|bool -> cterm list -> thm -> thm|, |
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249 where the \verb|bool| indicates goal mode, and the \verb|cterm list| the collection of assumptions to be discharged |
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250 simultaneously. |
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251 |
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252 \item \verb|Assumption.assume|~\isa{A} turns proposition \isa{A} into a raw assumption \isa{A\ {\isasymturnstile}\ A{\isacharprime}}, where the conclusion |
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253 \isa{A{\isacharprime}} is in HHF normal form. |
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254 |
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255 \item \verb|Assumption.add_assms|~\isa{r\ As} augments the context |
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256 by assumptions \isa{As} with export rule \isa{r}. The |
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257 resulting facts are hypothetical theorems as produced by the raw |
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258 \verb|Assumption.assume|. |
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259 |
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260 \item \verb|Assumption.add_assumes|~\isa{As} is a special case of |
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261 \verb|Assumption.add_assms| where the export rule performs \isa{{\isasymLongrightarrow}{\isacharunderscore}intro} or \isa{{\isacharhash}{\isasymLongrightarrow}{\isacharunderscore}intro}, depending on goal mode. |
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262 |
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263 \item \verb|Assumption.export|~\isa{is{\isacharunderscore}goal\ inner\ outer\ thm} |
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264 exports result \isa{thm} from the the \isa{inner} context |
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265 back into the \isa{outer} one; \isa{is{\isacharunderscore}goal\ {\isacharequal}\ true} means |
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266 this is a goal context. The result is in HHF normal form. Note |
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267 that \verb|ProofContext.export| combines \verb|Variable.export| |
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268 and \verb|Assumption.export| in the canonical way. |
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269 |
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270 \end{description}% |
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271 \end{isamarkuptext}% |
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272 \isamarkuptrue% |
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273 % |
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274 \endisatagmlref |
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275 {\isafoldmlref}% |
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276 % |
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277 \isadelimmlref |
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278 % |
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279 \endisadelimmlref |
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280 % |
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281 \isamarkupsection{Results \label{sec:results}% |
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282 } |
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283 \isamarkuptrue% |
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284 % |
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285 \begin{isamarkuptext}% |
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286 Local results are established by monotonic reasoning from facts |
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287 within a context. This allows common combinations of theorems, |
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288 e.g.\ via \isa{{\isasymAnd}{\isacharslash}{\isasymLongrightarrow}} elimination, resolution rules, or equational |
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289 reasoning, see \secref{sec:thms}. Unaccounted context manipulations |
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290 should be avoided, notably raw \isa{{\isasymAnd}{\isacharslash}{\isasymLongrightarrow}} introduction or ad-hoc |
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291 references to free variables or assumptions not present in the proof |
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292 context. |
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293 |
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294 \medskip The \isa{SUBPROOF} combinator allows to structure a |
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295 tactical proof recursively by decomposing a selected sub-goal: |
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296 \isa{{\isacharparenleft}{\isasymAnd}x{\isachardot}\ A{\isacharparenleft}x{\isacharparenright}\ {\isasymLongrightarrow}\ B{\isacharparenleft}x{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}} is turned into \isa{B{\isacharparenleft}x{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}} |
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297 after fixing \isa{x} and assuming \isa{A{\isacharparenleft}x{\isacharparenright}}. This means |
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298 the tactic needs to solve the conclusion, but may use the premise as |
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299 a local fact, for locally fixed variables. |
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300 |
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301 The \isa{prove} operation provides an interface for structured |
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302 backwards reasoning under program control, with some explicit sanity |
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303 checks of the result. The goal context can be augmented by |
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304 additional fixed variables (cf.\ \secref{sec:variables}) and |
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305 assumptions (cf.\ \secref{sec:assumptions}), which will be available |
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306 as local facts during the proof and discharged into implications in |
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307 the result. Type and term variables are generalized as usual, |
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308 according to the context. |
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309 |
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310 The \isa{obtain} operation produces results by eliminating |
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311 existing facts by means of a given tactic. This acts like a dual |
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312 conclusion: the proof demonstrates that the context may be augmented |
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313 by certain fixed variables and assumptions. See also |
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314 \cite{isabelle-isar-ref} for the user-level \isa{{\isasymOBTAIN}} and |
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315 \isa{{\isasymGUESS}} elements. Final results, which may not refer to |
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316 the parameters in the conclusion, need to exported explicitly into |
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317 the original context.% |
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318 \end{isamarkuptext}% |
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319 \isamarkuptrue% |
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320 % |
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321 \isadelimmlref |
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322 % |
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323 \endisadelimmlref |
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324 % |
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325 \isatagmlref |
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326 % |
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327 \begin{isamarkuptext}% |
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328 \begin{mldecls} |
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329 \indexml{SUBPROOF}\verb|SUBPROOF: ({context: Proof.context, schematics: ctyp list * cterm list,|\isasep\isanewline% |
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330 \verb| params: cterm list, asms: cterm list, concl: cterm,|\isasep\isanewline% |
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331 \verb| prems: thm list} -> tactic) -> Proof.context -> int -> tactic| \\ |
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332 \end{mldecls} |
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333 \begin{mldecls} |
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334 \indexml{Goal.prove}\verb|Goal.prove: Proof.context -> string list -> term list -> term ->|\isasep\isanewline% |
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335 \verb| ({prems: thm list, context: Proof.context} -> tactic) -> thm| \\ |
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336 \indexml{Goal.prove\_multi}\verb|Goal.prove_multi: Proof.context -> string list -> term list -> term list ->|\isasep\isanewline% |
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337 \verb| ({prems: thm list, context: Proof.context} -> tactic) -> thm list| \\ |
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338 \end{mldecls} |
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339 \begin{mldecls} |
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340 \indexml{Obtain.result}\verb|Obtain.result: (Proof.context -> tactic) ->|\isasep\isanewline% |
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341 \verb| thm list -> Proof.context -> (cterm list * thm list) * Proof.context| \\ |
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342 \end{mldecls} |
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343 |
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344 \begin{description} |
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345 |
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346 \item \verb|SUBPROOF|~\isa{tac} decomposes the structure of a |
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347 particular sub-goal, producing an extended context and a reduced |
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348 goal, which needs to be solved by the given tactic. All schematic |
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349 parameters of the goal are imported into the context as fixed ones, |
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350 which may not be instantiated in the sub-proof. |
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351 |
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352 \item \verb|Goal.prove|~\isa{ctxt\ xs\ As\ C\ tac} states goal \isa{C} in the context augmented by fixed variables \isa{xs} and |
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353 assumptions \isa{As}, and applies tactic \isa{tac} to solve |
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354 it. The latter may depend on the local assumptions being presented |
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355 as facts. The result is in HHF normal form. |
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356 |
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357 \item \verb|Goal.prove_multi| is simular to \verb|Goal.prove|, but |
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358 states several conclusions simultaneously. The goal is encoded by |
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359 means of Pure conjunction; \verb|Goal.conjunction_tac| will turn this |
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360 into a collection of individual subgoals. |
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361 |
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362 \item \verb|Obtain.result|~\isa{tac\ thms\ ctxt} eliminates the |
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363 given facts using a tactic, which results in additional fixed |
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364 variables and assumptions in the context. Final results need to be |
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365 exported explicitly. |
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366 |
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367 \end{description}% |
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368 \end{isamarkuptext}% |
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369 \isamarkuptrue% |
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370 % |
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371 \endisatagmlref |
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372 {\isafoldmlref}% |
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373 % |
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374 \isadelimmlref |
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375 % |
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376 \endisadelimmlref |
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377 % |
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378 \isadelimtheory |
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379 % |
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380 \endisadelimtheory |
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381 % |
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382 \isatagtheory |
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383 \isacommand{end}\isamarkupfalse% |
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384 % |
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385 \endisatagtheory |
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386 {\isafoldtheory}% |
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387 % |
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388 \isadelimtheory |
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389 % |
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390 \endisadelimtheory |
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391 \isanewline |
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392 \end{isabellebody}% |
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393 %%% Local Variables: |
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394 %%% mode: latex |
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395 %%% TeX-master: "root" |
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396 %%% End: |
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