doc-src/Logics/logics.toc
author wenzelm
Mon Nov 22 11:28:25 1993 +0100 (1993-11-22)
changeset 136 a9015b16a0e5
parent 104 d8205bb279a7
child 359 b5a2e9503a7a
permissions -rw-r--r--
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     1 \contentsline {chapter}{\numberline {1}Introduction}{1}
     2 \contentsline {section}{\numberline {1.1}Syntax definitions}{1}
     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}{16}
    13 \contentsline {subsection}{Deriving the introduction rule}{17}
    14 \contentsline {subsection}{Deriving the elimination rule}{17}
    15 \contentsline {subsection}{Using the derived rules}{18}
    16 \contentsline {subsection}{Derived rules versus definitions}{20}
    17 \contentsline {chapter}{\numberline {3}Zermelo-Fraenkel set theory}{23}
    18 \contentsline {section}{\numberline {3.1}Which version of axiomatic set theory?}{23}
    19 \contentsline {section}{\numberline {3.2}The syntax of set theory}{24}
    20 \contentsline {section}{\numberline {3.3}Binding operators}{26}
    21 \contentsline {section}{\numberline {3.4}The Zermelo-Fraenkel axioms}{28}
    22 \contentsline {section}{\numberline {3.5}From basic lemmas to function spaces}{33}
    23 \contentsline {subsection}{Fundamental lemmas}{34}
    24 \contentsline {subsection}{Unordered pairs and finite sets}{34}
    25 \contentsline {subsection}{Subset and lattice properties}{37}
    26 \contentsline {subsection}{Ordered pairs}{37}
    27 \contentsline {subsection}{Relations}{37}
    28 \contentsline {subsection}{Functions}{38}
    29 \contentsline {section}{\numberline {3.6}Further developments}{41}
    30 \contentsline {section}{\numberline {3.7}Simplification rules}{49}
    31 \contentsline {section}{\numberline {3.8}The examples directory}{49}
    32 \contentsline {section}{\numberline {3.9}A proof about powersets}{52}
    33 \contentsline {section}{\numberline {3.10}Monotonicity of the union operator}{54}
    34 \contentsline {section}{\numberline {3.11}Low-level reasoning about functions}{55}
    35 \contentsline {chapter}{\numberline {4}Higher-order logic}{58}
    36 \contentsline {section}{\numberline {4.1}Syntax}{58}
    37 \contentsline {subsection}{Types}{58}
    38 \contentsline {subsection}{Binders}{61}
    39 \contentsline {section}{\numberline {4.2}Rules of inference}{61}
    40 \contentsline {section}{\numberline {4.3}Generic packages}{65}
    41 \contentsline {section}{\numberline {4.4}A formulation of set theory}{66}
    42 \contentsline {subsection}{Syntax of set theory}{66}
    43 \contentsline {subsection}{Axioms and rules of set theory}{72}
    44 \contentsline {subsection}{Derived rules for sets}{72}
    45 \contentsline {section}{\numberline {4.5}Types}{72}
    46 \contentsline {subsection}{Product and sum types}{77}
    47 \contentsline {subsection}{The type of natural numbers, $nat$}{77}
    48 \contentsline {subsection}{The type constructor for lists, $\alpha \pcomma list$}{77}
    49 \contentsline {subsection}{The type constructor for lazy lists, $\alpha \pcomma llist$}{81}
    50 \contentsline {section}{\numberline {4.6}Classical proof procedures}{81}
    51 \contentsline {section}{\numberline {4.7}The examples directories}{81}
    52 \contentsline {section}{\numberline {4.8}Example: deriving the conjunction rules}{82}
    53 \contentsline {subsection}{The introduction rule}{82}
    54 \contentsline {subsection}{The elimination rule}{83}
    55 \contentsline {section}{\numberline {4.9}Example: Cantor's Theorem}{84}
    56 \contentsline {chapter}{\numberline {5}First-order sequent calculus}{87}
    57 \contentsline {section}{\numberline {5.1}Unification for lists}{87}
    58 \contentsline {section}{\numberline {5.2}Syntax and rules of inference}{88}
    59 \contentsline {section}{\numberline {5.3}Tactics for the cut rule}{88}
    60 \contentsline {section}{\numberline {5.4}Tactics for sequents}{93}
    61 \contentsline {section}{\numberline {5.5}Packaging sequent rules}{93}
    62 \contentsline {section}{\numberline {5.6}Proof procedures}{94}
    63 \contentsline {subsection}{Method A}{95}
    64 \contentsline {subsection}{Method B}{95}
    65 \contentsline {section}{\numberline {5.7}A simple example of classical reasoning}{95}
    66 \contentsline {section}{\numberline {5.8}A more complex proof}{97}
    67 \contentsline {chapter}{\numberline {6}Constructive Type Theory}{99}
    68 \contentsline {section}{\numberline {6.1}Syntax}{100}
    69 \contentsline {section}{\numberline {6.2}Rules of inference}{100}
    70 \contentsline {section}{\numberline {6.3}Rule lists}{105}
    71 \contentsline {section}{\numberline {6.4}Tactics for subgoal reordering}{108}
    72 \contentsline {section}{\numberline {6.5}Rewriting tactics}{109}
    73 \contentsline {section}{\numberline {6.6}Tactics for logical reasoning}{109}
    74 \contentsline {section}{\numberline {6.7}A theory of arithmetic}{110}
    75 \contentsline {section}{\numberline {6.8}The examples directory}{110}
    76 \contentsline {section}{\numberline {6.9}Example: type inference}{112}
    77 \contentsline {section}{\numberline {6.10}An example of logical reasoning}{113}
    78 \contentsline {section}{\numberline {6.11}Example: deriving a currying functional}{116}
    79 \contentsline {section}{\numberline {6.12}Example: proving the Axiom of Choice}{117}
    80 \contentsline {chapter}{\numberline {7}Defining Logics}{121}
    81 \contentsline {section}{\numberline {7.1}Precedence grammars}{121}
    82 \contentsline {section}{\numberline {7.2}Basic syntax}{122}
    83 \contentsline {subsection}{Logical types and default syntax}{123}
    84 \contentsline {subsection}{Lexical matters *}{124}
    85 \contentsline {subsection}{Inspecting syntax *}{124}
    86 \contentsline {section}{\numberline {7.3}Abstract syntax trees}{126}
    87 \contentsline {subsection}{Parse trees to asts}{128}
    88 \contentsline {subsection}{Asts to terms *}{129}
    89 \contentsline {subsection}{Printing of terms *}{129}
    90 \contentsline {section}{\numberline {7.4}Mixfix declarations}{130}
    91 \contentsline {subsection}{Infixes}{133}
    92 \contentsline {subsection}{Binders}{133}
    93 \contentsline {section}{\numberline {7.5}Syntactic translations (macros)}{134}
    94 \contentsline {subsection}{Specifying macros}{135}
    95 \contentsline {subsection}{Applying rules}{136}
    96 \contentsline {subsection}{Rewriting strategy}{138}
    97 \contentsline {subsection}{More examples}{138}
    98 \contentsline {section}{\numberline {7.6}Translation functions *}{141}
    99 \contentsline {subsection}{A simple example *}{142}
   100 \contentsline {section}{\numberline {7.7}Example: some minimal logics}{143}