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1 \contentsline {part}{\uppercase {i}\phspace {1em}Foundations}{1} |
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2 \contentsline {section}{\numberline {1}Formalizing logical syntax in Isabelle}{1} |
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3 \contentsline {subsection}{\numberline {1.1}Simple types and constants}{1} |
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4 \contentsline {subsection}{\numberline {1.2}Polymorphic types and constants}{3} |
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5 \contentsline {subsection}{\numberline {1.3}Higher types and quantifiers}{4} |
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6 \contentsline {section}{\numberline {2}Formalizing logical rules in Isabelle}{5} |
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7 \contentsline {subsection}{\numberline {2.1}Expressing propositional rules}{6} |
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8 \contentsline {subsection}{\numberline {2.2}Quantifier rules and substitution}{7} |
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9 \contentsline {subsection}{\numberline {2.3}Signatures and theories}{8} |
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10 \contentsline {section}{\numberline {3}Proof construction in Isabelle}{9} |
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11 \contentsline {subsection}{\numberline {3.1}Higher-order unification}{10} |
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12 \contentsline {subsection}{\numberline {3.2}Joining rules by resolution}{11} |
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13 \contentsline {subsection}{\numberline {3.3}Lifting a rule into a context}{13} |
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14 \contentsline {subsubsection}{Lifting over assumptions}{13} |
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15 \contentsline {subsubsection}{Lifting over parameters}{13} |
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16 \contentsline {section}{\numberline {4}Backward proof by resolution}{14} |
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17 \contentsline {subsection}{\numberline {4.1}Refinement by resolution}{15} |
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18 \contentsline {subsection}{\numberline {4.2}Proof by assumption}{15} |
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19 \contentsline {subsection}{\numberline {4.3}A propositional proof}{16} |
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20 \contentsline {subsection}{\numberline {4.4}A quantifier proof}{17} |
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21 \contentsline {subsection}{\numberline {4.5}Tactics and tacticals}{17} |
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22 \contentsline {section}{\numberline {5}Variations on resolution}{18} |
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23 \contentsline {subsection}{\numberline {5.1}Elim-resolution}{18} |
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24 \contentsline {subsection}{\numberline {5.2}Destruction rules}{20} |
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25 \contentsline {subsection}{\numberline {5.3}Deriving rules by resolution}{20} |
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26 \contentsline {part}{\uppercase {ii}\phspace {1em}Getting started with Isabelle}{22} |
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27 \contentsline {section}{\numberline {6}Forward proof}{22} |
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28 \contentsline {subsection}{\numberline {6.1}Lexical matters}{22} |
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29 \contentsline {subsection}{\numberline {6.2}Syntax of types and terms}{23} |
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30 \contentsline {subsection}{\numberline {6.3}Basic operations on theorems}{24} |
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31 \contentsline {subsection}{\numberline {6.4}Flex-flex equations}{26} |
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32 \contentsline {section}{\numberline {7}Backward proof}{27} |
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33 \contentsline {subsection}{\numberline {7.1}The basic tactics}{27} |
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34 \contentsline {subsection}{\numberline {7.2}Commands for backward proof}{28} |
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35 \contentsline {subsection}{\numberline {7.3}A trivial example in propositional logic}{28} |
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36 \contentsline {subsection}{\numberline {7.4}Proving a distributive law}{30} |
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37 \contentsline {section}{\numberline {8}Quantifier reasoning}{31} |
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38 \contentsline {subsection}{\numberline {8.1}Two quantifier proofs, successful and not}{31} |
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39 \contentsline {subsubsection}{The successful proof}{31} |
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40 \contentsline {subsubsection}{The unsuccessful proof}{32} |
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41 \contentsline {subsection}{\numberline {8.2}Nested quantifiers}{33} |
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42 \contentsline {subsubsection}{The wrong approach}{33} |
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43 \contentsline {subsubsection}{The right approach}{34} |
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44 \contentsline {subsubsection}{A one-step proof using tacticals}{35} |
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45 \contentsline {subsection}{\numberline {8.3}A realistic quantifier proof}{35} |
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46 \contentsline {subsection}{\numberline {8.4}The classical reasoning package}{36} |
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47 \contentsline {part}{\uppercase {iii}\phspace {1em}Advanced methods}{38} |
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48 \contentsline {section}{\numberline {9}Deriving rules in Isabelle}{38} |
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49 \contentsline {subsection}{\numberline {9.1}Deriving a rule using tactics}{38} |
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50 \contentsline {subsection}{\numberline {9.2}Definitions and derived rules}{40} |
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51 \contentsline {subsubsection}{Deriving the introduction rule}{41} |
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52 \contentsline {subsubsection}{Deriving the elimination rule}{42} |
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53 \contentsline {section}{\numberline {10}Defining theories}{44} |
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54 \contentsline {subsection}{\numberline {10.1}Declaring constants and rules}{45} |
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55 \contentsline {subsection}{\numberline {10.2}Declaring type constructors}{46} |
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56 \contentsline {subsection}{\numberline {10.3}Infixes and Mixfixes}{47} |
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57 \contentsline {subsection}{\numberline {10.4}Overloading}{48} |
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58 \contentsline {subsection}{\numberline {10.5}Extending first-order logic with the natural numbers}{50} |
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59 \contentsline {subsection}{\numberline {10.6}Declaring the theory to Isabelle}{51} |
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60 \contentsline {section}{\numberline {11}Refinement with explicit instantiation}{52} |
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61 \contentsline {subsection}{\numberline {11.1}A simple proof by induction}{52} |
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62 \contentsline {subsection}{\numberline {11.2}An example of ambiguity in {\ptt resolve_tac}}{53} |
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63 \contentsline {subsection}{\numberline {11.3}Proving that addition is associative}{54} |
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64 \contentsline {section}{\numberline {12}A {\psc Prolog} interpreter}{55} |
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65 \contentsline {subsection}{\numberline {12.1}Simple executions}{56} |
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66 \contentsline {subsection}{\numberline {12.2}Backtracking}{57} |
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67 \contentsline {subsection}{\numberline {12.3}Depth-first search}{58} |