doc-src/Logics/logics.toc
author lcp
Tue May 03 18:38:28 1994 +0200 (1994-05-03)
changeset 359 b5a2e9503a7a
parent 136 a9015b16a0e5
child 465 d4bf81734dfe
permissions -rw-r--r--
final Springer version
     1 \contentsline {chapter}{\numberline {1}Basic Concepts}{1}
     2 \contentsline {section}{\numberline {1.1}Syntax definitions}{2}
     3 \contentsline {section}{\numberline {1.2}Proof procedures}{3}
     4 \contentsline {chapter}{\numberline {2}First-Order Logic}{4}
     5 \contentsline {section}{\numberline {2.1}Syntax and rules of inference}{4}
     6 \contentsline {section}{\numberline {2.2}Generic packages}{8}
     7 \contentsline {section}{\numberline {2.3}Intuitionistic proof procedures}{8}
     8 \contentsline {section}{\numberline {2.4}Classical proof procedures}{10}
     9 \contentsline {section}{\numberline {2.5}An intuitionistic example}{11}
    10 \contentsline {section}{\numberline {2.6}An example of intuitionistic negation}{12}
    11 \contentsline {section}{\numberline {2.7}A classical example}{14}
    12 \contentsline {section}{\numberline {2.8}Derived rules and the classical tactics}{15}
    13 \contentsline {subsection}{Deriving the introduction rule}{16}
    14 \contentsline {subsection}{Deriving the elimination rule}{17}
    15 \contentsline {subsection}{Using the derived rules}{17}
    16 \contentsline {subsection}{Derived rules versus definitions}{19}
    17 \contentsline {chapter}{\numberline {3}Zermelo-Fraenkel Set Theory}{22}
    18 \contentsline {section}{\numberline {3.1}Which version of axiomatic set theory?}{22}
    19 \contentsline {section}{\numberline {3.2}The syntax of set theory}{23}
    20 \contentsline {section}{\numberline {3.3}Binding operators}{25}
    21 \contentsline {section}{\numberline {3.4}The Zermelo-Fraenkel axioms}{27}
    22 \contentsline {section}{\numberline {3.5}From basic lemmas to function spaces}{30}
    23 \contentsline {subsection}{Fundamental lemmas}{30}
    24 \contentsline {subsection}{Unordered pairs and finite sets}{32}
    25 \contentsline {subsection}{Subset and lattice properties}{32}
    26 \contentsline {subsection}{Ordered pairs}{36}
    27 \contentsline {subsection}{Relations}{36}
    28 \contentsline {subsection}{Functions}{37}
    29 \contentsline {section}{\numberline {3.6}Further developments}{38}
    30 \contentsline {section}{\numberline {3.7}Simplification rules}{47}
    31 \contentsline {section}{\numberline {3.8}The examples directory}{47}
    32 \contentsline {section}{\numberline {3.9}A proof about powersets}{48}
    33 \contentsline {section}{\numberline {3.10}Monotonicity of the union operator}{51}
    34 \contentsline {section}{\numberline {3.11}Low-level reasoning about functions}{52}
    35 \contentsline {chapter}{\numberline {4}Higher-Order Logic}{55}
    36 \contentsline {section}{\numberline {4.1}Syntax}{55}
    37 \contentsline {subsection}{Types}{57}
    38 \contentsline {subsection}{Binders}{58}
    39 \contentsline {subsection}{The {\ptt let} and {\ptt case} constructions}{58}
    40 \contentsline {section}{\numberline {4.2}Rules of inference}{58}
    41 \contentsline {section}{\numberline {4.3}A formulation of set theory}{60}
    42 \contentsline {subsection}{Syntax of set theory}{65}
    43 \contentsline {subsection}{Axioms and rules of set theory}{69}
    44 \contentsline {section}{\numberline {4.4}Generic packages and classical reasoning}{71}
    45 \contentsline {section}{\numberline {4.5}Types}{73}
    46 \contentsline {subsection}{Product and sum types}{73}
    47 \contentsline {subsection}{The type of natural numbers, {\ptt nat}}{73}
    48 \contentsline {subsection}{The type constructor for lists, {\ptt list}}{76}
    49 \contentsline {subsection}{The type constructor for lazy lists, {\ptt llist}}{76}
    50 \contentsline {section}{\numberline {4.6}The examples directories}{79}
    51 \contentsline {section}{\numberline {4.7}Example: Cantor's Theorem}{80}
    52 \contentsline {chapter}{\numberline {5}First-Order Sequent Calculus}{82}
    53 \contentsline {section}{\numberline {5.1}Unification for lists}{82}
    54 \contentsline {section}{\numberline {5.2}Syntax and rules of inference}{84}
    55 \contentsline {section}{\numberline {5.3}Tactics for the cut rule}{86}
    56 \contentsline {section}{\numberline {5.4}Tactics for sequents}{87}
    57 \contentsline {section}{\numberline {5.5}Packaging sequent rules}{88}
    58 \contentsline {section}{\numberline {5.6}Proof procedures}{88}
    59 \contentsline {subsection}{Method A}{89}
    60 \contentsline {subsection}{Method B}{89}
    61 \contentsline {section}{\numberline {5.7}A simple example of classical reasoning}{90}
    62 \contentsline {section}{\numberline {5.8}A more complex proof}{91}
    63 \contentsline {chapter}{\numberline {6}Constructive Type Theory}{93}
    64 \contentsline {section}{\numberline {6.1}Syntax}{95}
    65 \contentsline {section}{\numberline {6.2}Rules of inference}{95}
    66 \contentsline {section}{\numberline {6.3}Rule lists}{101}
    67 \contentsline {section}{\numberline {6.4}Tactics for subgoal reordering}{101}
    68 \contentsline {section}{\numberline {6.5}Rewriting tactics}{102}
    69 \contentsline {section}{\numberline {6.6}Tactics for logical reasoning}{103}
    70 \contentsline {section}{\numberline {6.7}A theory of arithmetic}{105}
    71 \contentsline {section}{\numberline {6.8}The examples directory}{105}
    72 \contentsline {section}{\numberline {6.9}Example: type inference}{105}
    73 \contentsline {section}{\numberline {6.10}An example of logical reasoning}{107}
    74 \contentsline {section}{\numberline {6.11}Example: deriving a currying functional}{110}
    75 \contentsline {section}{\numberline {6.12}Example: proving the Axiom of Choice}{111}