\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}{Deriving the introduction rule}{16}
\contentsline {subsection}{Deriving the elimination rule}{17}
\contentsline {subsection}{Using the derived rules}{17}
\contentsline {subsection}{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}{Fundamental lemmas}{30}
\contentsline {subsection}{Unordered pairs and finite sets}{32}
\contentsline {subsection}{Subset and lattice properties}{32}
\contentsline {subsection}{Ordered pairs}{36}
\contentsline {subsection}{Relations}{36}
\contentsline {subsection}{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}{Types}{57}
\contentsline {subsection}{Binders}{58}
\contentsline {subsection}{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}{Syntax of set theory}{65}
\contentsline {subsection}{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}{Product and sum types}{73}
\contentsline {subsection}{The type of natural numbers, {\ptt nat}}{73}
\contentsline {subsection}{The type constructor for lists, {\ptt list}}{76}
\contentsline {subsection}{The type constructor for lazy lists, {\ptt llist}}{76}
\contentsline {section}{\numberline {4.6}The examples directories}{79}
\contentsline {section}{\numberline {4.7}Example: Cantor's Theorem}{80}
\contentsline {chapter}{\numberline {5}First-Order Sequent Calculus}{82}
\contentsline {section}{\numberline {5.1}Unification for lists}{82}
\contentsline {section}{\numberline {5.2}Syntax and rules of inference}{84}
\contentsline {section}{\numberline {5.3}Tactics for the cut rule}{86}
\contentsline {section}{\numberline {5.4}Tactics for sequents}{87}
\contentsline {section}{\numberline {5.5}Packaging sequent rules}{88}
\contentsline {section}{\numberline {5.6}Proof procedures}{88}
\contentsline {subsection}{Method A}{89}
\contentsline {subsection}{Method B}{89}
\contentsline {section}{\numberline {5.7}A simple example of classical reasoning}{90}
\contentsline {section}{\numberline {5.8}A more complex proof}{91}
\contentsline {chapter}{\numberline {6}Constructive Type Theory}{93}
\contentsline {section}{\numberline {6.1}Syntax}{95}
\contentsline {section}{\numberline {6.2}Rules of inference}{95}
\contentsline {section}{\numberline {6.3}Rule lists}{101}
\contentsline {section}{\numberline {6.4}Tactics for subgoal reordering}{101}
\contentsline {section}{\numberline {6.5}Rewriting tactics}{102}
\contentsline {section}{\numberline {6.6}Tactics for logical reasoning}{103}
\contentsline {section}{\numberline {6.7}A theory of arithmetic}{105}
\contentsline {section}{\numberline {6.8}The examples directory}{105}
\contentsline {section}{\numberline {6.9}Example: type inference}{105}
\contentsline {section}{\numberline {6.10}An example of logical reasoning}{107}
\contentsline {section}{\numberline {6.11}Example: deriving a currying functional}{110}
\contentsline {section}{\numberline {6.12}Example: proving the Axiom of Choice}{111}