doc-src/Logics/logics.toc
author lcp
Wed Nov 10 05:00:57 1993 +0100 (1993-11-10)
changeset 104 d8205bb279a7
child 136 a9015b16a0e5
permissions -rw-r--r--
Initial revision
lcp@104
     1
\contentsline {chapter}{\numberline {1}Introduction}{1}
lcp@104
     2
\contentsline {section}{\numberline {1.1}Syntax definitions}{1}
lcp@104
     3
\contentsline {section}{\numberline {1.2}Proof procedures}{3}
lcp@104
     4
\contentsline {chapter}{\numberline {2}First-order logic}{4}
lcp@104
     5
\contentsline {section}{\numberline {2.1}Syntax and rules of inference}{4}
lcp@104
     6
\contentsline {section}{\numberline {2.2}Generic packages}{8}
lcp@104
     7
\contentsline {section}{\numberline {2.3}Intuitionistic proof procedures}{8}
lcp@104
     8
\contentsline {section}{\numberline {2.4}Classical proof procedures}{10}
lcp@104
     9
\contentsline {section}{\numberline {2.5}An intuitionistic example}{11}
lcp@104
    10
\contentsline {section}{\numberline {2.6}An example of intuitionistic negation}{12}
lcp@104
    11
\contentsline {section}{\numberline {2.7}A classical example}{14}
lcp@104
    12
\contentsline {section}{\numberline {2.8}Derived rules and the classical tactics}{16}
lcp@104
    13
\contentsline {subsection}{Deriving the introduction rule}{17}
lcp@104
    14
\contentsline {subsection}{Deriving the elimination rule}{17}
lcp@104
    15
\contentsline {subsection}{Using the derived rules}{18}
lcp@104
    16
\contentsline {subsection}{Derived rules versus definitions}{20}
lcp@104
    17
\contentsline {chapter}{\numberline {3}Zermelo-Fraenkel set theory}{23}
lcp@104
    18
\contentsline {section}{\numberline {3.1}Which version of axiomatic set theory?}{23}
lcp@104
    19
\contentsline {section}{\numberline {3.2}The syntax of set theory}{25}
lcp@104
    20
\contentsline {section}{\numberline {3.3}Binding operators}{25}
lcp@104
    21
\contentsline {section}{\numberline {3.4}The Zermelo-Fraenkel axioms}{28}
lcp@104
    22
\contentsline {section}{\numberline {3.5}From basic lemmas to function spaces}{33}
lcp@104
    23
\contentsline {subsection}{Fundamental lemmas}{33}
lcp@104
    24
\contentsline {subsection}{Unordered pairs and finite sets}{36}
lcp@104
    25
\contentsline {subsection}{Subset and lattice properties}{36}
lcp@104
    26
\contentsline {subsection}{Ordered pairs}{37}
lcp@104
    27
\contentsline {subsection}{Relations}{37}
lcp@104
    28
\contentsline {subsection}{Functions}{40}
lcp@104
    29
\contentsline {section}{\numberline {3.6}Further developments}{40}
lcp@104
    30
\contentsline {section}{\numberline {3.7}Simplification rules}{47}
lcp@104
    31
\contentsline {section}{\numberline {3.8}The examples directory}{48}
lcp@104
    32
\contentsline {section}{\numberline {3.9}A proof about powersets}{49}
lcp@104
    33
\contentsline {section}{\numberline {3.10}Monotonicity of the union operator}{51}
lcp@104
    34
\contentsline {section}{\numberline {3.11}Low-level reasoning about functions}{52}
lcp@104
    35
\contentsline {chapter}{\numberline {4}Higher-order logic}{55}
lcp@104
    36
\contentsline {section}{\numberline {4.1}Syntax}{55}
lcp@104
    37
\contentsline {subsection}{Types}{55}
lcp@104
    38
\contentsline {subsection}{Binders}{58}
lcp@104
    39
\contentsline {section}{\numberline {4.2}Rules of inference}{58}
lcp@104
    40
\contentsline {section}{\numberline {4.3}Generic packages}{62}
lcp@104
    41
\contentsline {section}{\numberline {4.4}A formulation of set theory}{63}
lcp@104
    42
\contentsline {subsection}{Syntax of set theory}{63}
lcp@104
    43
\contentsline {subsection}{Axioms and rules of set theory}{69}
lcp@104
    44
\contentsline {subsection}{Derived rules for sets}{69}
lcp@104
    45
\contentsline {section}{\numberline {4.5}Types}{69}
lcp@104
    46
\contentsline {subsection}{Product and sum types}{74}
lcp@104
    47
\contentsline {subsection}{The type of natural numbers, $nat$}{74}
lcp@104
    48
\contentsline {subsection}{The type constructor for lists, $\alpha \pcomma list$}{74}
lcp@104
    49
\contentsline {subsection}{The type constructor for lazy lists, $\alpha \pcomma llist$}{78}
lcp@104
    50
\contentsline {section}{\numberline {4.6}Classical proof procedures}{78}
lcp@104
    51
\contentsline {section}{\numberline {4.7}The examples directory}{78}
lcp@104
    52
\contentsline {section}{\numberline {4.8}Example: deriving the conjunction rules}{79}
lcp@104
    53
\contentsline {subsection}{The introduction rule}{79}
lcp@104
    54
\contentsline {subsection}{The elimination rule}{80}
lcp@104
    55
\contentsline {section}{\numberline {4.9}Example: Cantor's Theorem}{81}
lcp@104
    56
\contentsline {chapter}{\numberline {5}First-order sequent calculus}{83}
lcp@104
    57
\contentsline {section}{\numberline {5.1}Unification for lists}{83}
lcp@104
    58
\contentsline {section}{\numberline {5.2}Syntax and rules of inference}{84}
lcp@104
    59
\contentsline {section}{\numberline {5.3}Tactics for the cut rule}{84}
lcp@104
    60
\contentsline {section}{\numberline {5.4}Tactics for sequents}{88}
lcp@104
    61
\contentsline {section}{\numberline {5.5}Packaging sequent rules}{89}
lcp@104
    62
\contentsline {section}{\numberline {5.6}Proof procedures}{89}
lcp@104
    63
\contentsline {subsection}{Method A}{90}
lcp@104
    64
\contentsline {subsection}{Method B}{90}
lcp@104
    65
\contentsline {section}{\numberline {5.7}A simple example of classical reasoning}{91}
lcp@104
    66
\contentsline {section}{\numberline {5.8}A more complex proof}{92}
lcp@104
    67
\contentsline {chapter}{\numberline {6}Constructive Type Theory}{95}
lcp@104
    68
\contentsline {section}{\numberline {6.1}Syntax}{96}
lcp@104
    69
\contentsline {section}{\numberline {6.2}Rules of inference}{96}
lcp@104
    70
\contentsline {section}{\numberline {6.3}Rule lists}{101}
lcp@104
    71
\contentsline {section}{\numberline {6.4}Tactics for subgoal reordering}{104}
lcp@104
    72
\contentsline {section}{\numberline {6.5}Rewriting tactics}{105}
lcp@104
    73
\contentsline {section}{\numberline {6.6}Tactics for logical reasoning}{105}
lcp@104
    74
\contentsline {section}{\numberline {6.7}A theory of arithmetic}{106}
lcp@104
    75
\contentsline {section}{\numberline {6.8}The examples directory}{106}
lcp@104
    76
\contentsline {section}{\numberline {6.9}Example: type inference}{108}
lcp@104
    77
\contentsline {section}{\numberline {6.10}An example of logical reasoning}{109}
lcp@104
    78
\contentsline {section}{\numberline {6.11}Example: deriving a currying functional}{112}
lcp@104
    79
\contentsline {section}{\numberline {6.12}Example: proving the Axiom of Choice}{113}
lcp@104
    80
\contentsline {chapter}{\numberline {7}Defining Logics}{118}
lcp@104
    81
\contentsline {section}{\numberline {7.1}Precedence grammars}{118}
lcp@104
    82
\contentsline {section}{\numberline {7.2}Basic syntax}{119}
lcp@104
    83
\contentsline {subsection}{Logical types and default syntax}{120}
lcp@104
    84
\contentsline {subsection}{Lexical matters *}{121}
lcp@104
    85
\contentsline {subsection}{Inspecting syntax *}{121}
lcp@104
    86
\contentsline {section}{\numberline {7.3}Abstract syntax trees}{123}
lcp@104
    87
\contentsline {subsection}{Parse trees to asts}{125}
lcp@104
    88
\contentsline {subsection}{Asts to terms *}{126}
lcp@104
    89
\contentsline {subsection}{Printing of terms *}{126}
lcp@104
    90
\contentsline {section}{\numberline {7.4}Mixfix declarations}{128}
lcp@104
    91
\contentsline {subsection}{Infixes}{130}
lcp@104
    92
\contentsline {subsection}{Binders}{130}
lcp@104
    93
\contentsline {section}{\numberline {7.5}Syntactic translations (macros)}{131}
lcp@104
    94
\contentsline {subsection}{Specifying macros}{132}
lcp@104
    95
\contentsline {subsection}{Applying rules}{133}
lcp@104
    96
\contentsline {subsection}{Rewriting strategy}{135}
lcp@104
    97
\contentsline {subsection}{More examples}{135}
lcp@104
    98
\contentsline {section}{\numberline {7.6}Translation functions *}{138}
lcp@104
    99
\contentsline {subsection}{A simple example *}{139}
lcp@104
   100
\contentsline {section}{\numberline {7.7}Example: some minimal logics}{140}