\contentsline {chapter}{\numberline {1}Basic Concepts}{1}
\contentsline {section}{\numberline {1.1}Syntax definitions}{2}
\contentsline {section}{\numberline {1.2}Proof procedures}{3}
\contentsline {chapter}{\numberline {2}First-Order Logic}{4}
\contentsline {section}{\numberline {2.1}Syntax and rules of inference}{4}
\contentsline {section}{\numberline {2.2}Generic packages}{8}
\contentsline {section}{\numberline {2.3}Intuitionistic proof procedures}{8}
\contentsline {section}{\numberline {2.4}Classical proof procedures}{10}
\contentsline {section}{\numberline {2.5}An intuitionistic example}{11}
\contentsline {section}{\numberline {2.6}An example of intuitionistic negation}{12}
\contentsline {section}{\numberline {2.7}A classical example}{14}
\contentsline {section}{\numberline {2.8}Derived rules and the classical tactics}{15}
\contentsline {subsection}{\numberline {2.8.1}Deriving the introduction rule}{16}
\contentsline {subsection}{\numberline {2.8.2}Deriving the elimination rule}{17}
\contentsline {subsection}{\numberline {2.8.3}Using the derived rules}{17}
\contentsline {subsection}{\numberline {2.8.4}Derived rules versus definitions}{19}
\contentsline {chapter}{\numberline {3}Zermelo-Fraenkel Set Theory}{22}
\contentsline {section}{\numberline {3.1}Which version of axiomatic set theory?}{22}
\contentsline {section}{\numberline {3.2}The syntax of set theory}{23}
\contentsline {section}{\numberline {3.3}Binding operators}{25}
\contentsline {section}{\numberline {3.4}The Zermelo-Fraenkel axioms}{27}
\contentsline {section}{\numberline {3.5}From basic lemmas to function spaces}{30}
\contentsline {subsection}{\numberline {3.5.1}Fundamental lemmas}{30}
\contentsline {subsection}{\numberline {3.5.2}Unordered pairs and finite sets}{32}
\contentsline {subsection}{\numberline {3.5.3}Subset and lattice properties}{32}
\contentsline {subsection}{\numberline {3.5.4}Ordered pairs}{36}
\contentsline {subsection}{\numberline {3.5.5}Relations}{36}
\contentsline {subsection}{\numberline {3.5.6}Functions}{37}
\contentsline {section}{\numberline {3.6}Further developments}{38}
\contentsline {section}{\numberline {3.7}Simplification rules}{47}
\contentsline {section}{\numberline {3.8}The examples directory}{47}
\contentsline {section}{\numberline {3.9}A proof about powersets}{48}
\contentsline {section}{\numberline {3.10}Monotonicity of the union operator}{51}
\contentsline {section}{\numberline {3.11}Low-level reasoning about functions}{52}
\contentsline {chapter}{\numberline {4}Higher-Order Logic}{55}
\contentsline {section}{\numberline {4.1}Syntax}{55}
\contentsline {subsection}{\numberline {4.1.1}Types}{57}
\contentsline {subsection}{\numberline {4.1.2}Binders}{58}
\contentsline {subsection}{\numberline {4.1.3}The {\ptt let} and {\ptt case} constructions}{58}
\contentsline {section}{\numberline {4.2}Rules of inference}{58}
\contentsline {section}{\numberline {4.3}A formulation of set theory}{60}
\contentsline {subsection}{\numberline {4.3.1}Syntax of set theory}{65}
\contentsline {subsection}{\numberline {4.3.2}Axioms and rules of set theory}{69}
\contentsline {section}{\numberline {4.4}Generic packages and classical reasoning}{71}
\contentsline {section}{\numberline {4.5}Types}{73}
\contentsline {subsection}{\numberline {4.5.1}Product and sum types}{73}
\contentsline {subsection}{\numberline {4.5.2}The type of natural numbers, {\ptt nat}}{73}
\contentsline {subsection}{\numberline {4.5.3}The type constructor for lists, {\ptt list}}{76}
\contentsline {subsection}{\numberline {4.5.4}The type constructor for lazy lists, {\ptt llist}}{76}
\contentsline {section}{\numberline {4.6}Datatype declarations}{79}
\contentsline {subsection}{\numberline {4.6.1}Foundations}{79}
\contentsline {subsection}{\numberline {4.6.2}Defining datatypes}{80}
\contentsline {subsection}{\numberline {4.6.3}Examples}{82}
\contentsline {subsubsection}{The datatype $\alpha \penalty \@M \ list$}{82}
\contentsline {subsubsection}{The datatype $\alpha \penalty \@M \ list$ with mixfix syntax}{83}
\contentsline {subsubsection}{Defining functions on datatypes}{83}
\contentsline {subsubsection}{A datatype for weekdays}{84}
\contentsline {section}{\numberline {4.7}The examples directories}{84}
\contentsline {section}{\numberline {4.8}Example: Cantor's Theorem}{85}
\contentsline {chapter}{\numberline {5}First-Order Sequent Calculus}{88}
\contentsline {section}{\numberline {5.1}Unification for lists}{88}
\contentsline {section}{\numberline {5.2}Syntax and rules of inference}{90}
\contentsline {section}{\numberline {5.3}Tactics for the cut rule}{92}
\contentsline {section}{\numberline {5.4}Tactics for sequents}{93}
\contentsline {section}{\numberline {5.5}Packaging sequent rules}{94}
\contentsline {section}{\numberline {5.6}Proof procedures}{94}
\contentsline {subsection}{\numberline {5.6.1}Method A}{95}
\contentsline {subsection}{\numberline {5.6.2}Method B}{95}
\contentsline {section}{\numberline {5.7}A simple example of classical reasoning}{96}
\contentsline {section}{\numberline {5.8}A more complex proof}{97}
\contentsline {chapter}{\numberline {6}Constructive Type Theory}{99}
\contentsline {section}{\numberline {6.1}Syntax}{101}
\contentsline {section}{\numberline {6.2}Rules of inference}{101}
\contentsline {section}{\numberline {6.3}Rule lists}{107}
\contentsline {section}{\numberline {6.4}Tactics for subgoal reordering}{107}
\contentsline {section}{\numberline {6.5}Rewriting tactics}{108}
\contentsline {section}{\numberline {6.6}Tactics for logical reasoning}{109}
\contentsline {section}{\numberline {6.7}A theory of arithmetic}{111}
\contentsline {section}{\numberline {6.8}The examples directory}{111}
\contentsline {section}{\numberline {6.9}Example: type inference}{111}
\contentsline {section}{\numberline {6.10}An example of logical reasoning}{113}
\contentsline {section}{\numberline {6.11}Example: deriving a currying functional}{116}
\contentsline {section}{\numberline {6.12}Example: proving the Axiom of Choice}{117}