doc-src/TutorialI/tutorial.ind

intro! and elim! rules;

\begin{theindex}

  \item \ttall, \bold{195}
  \item \texttt{?}, \bold{195}
  \item \isasymuniqex, \bold{195}
  \item \ttuniquex, \bold{195}
  \item {\texttt {\&}}, \bold{195}
  \item \verb$~$, \bold{195}
  \item \verb$~=$, \bold{195}
  \item \ttor, \bold{195}
  \item \texttt{[]}, \bold{9}
  \item \texttt{\#}, \bold{9}
  \item \texttt{\at}, \bold{10}, 195
  \item \isasymnotin, \bold{195}
  \item \verb$~:$, \bold{195}
  \item \isasymInter, \bold{195}
  \item \isasymUnion, \bold{195}
  \item \isasyminverse, \bold{195}
  \item \verb$^-1$, \bold{195}
  \item \isactrlsup{\isacharasterisk}, \bold{195}
  \item \verb$^$\texttt{*}, \bold{195}
  \item \isasymAnd, \bold{12}, \bold{195}
  \item \ttAnd, \bold{195}
  \item \isasymrightleftharpoons, 26
  \item \isasymrightharpoonup, 26
  \item \isasymleftharpoondown, 26
  \item \emph {$\Rightarrow $}, \bold{5}
  \item \ttlbr, \bold{195}
  \item \ttrbr, \bold{195}
  \item \texttt {\%}, \bold{195}
  \item \texttt {;}, \bold{7}
  \item \isa {()} (constant), 24
  \item \isa {+} (tactical), 89
  \item \isa {<*lex*>}, \see{lexicographic product}{1}
  \item \isa {?} (tactical), 89
  \item \texttt{|} (tactical), 89

  \indexspace

  \item \isa {0} (constant), 22, 23, 141
  \item \isa {1} (symbol), 141
  \item \isa {2} (symbol), 141

  \indexspace

  \item abandoning a proof, \bold{13}
  \item abandoning a theory, \bold{16}
  \item \isa {abs} (constant), 143
  \item \texttt {abs}, \bold{195}
  \item absolute value, 143
  \item \isa {add} (modifier), 29
  \item \isa {add_ac} (theorems), 142
  \item \isa {add_assoc} (theorem), \bold{142}
  \item \isa {add_commute} (theorem), \bold{142}
  \item \isa {add_mult_distrib} (theorem), \bold{141}
  \item \texttt {ALL}, \bold{195}
  \item \isa {All} (constant), 99
  \item \isa {allE} (theorem), \bold{71}
  \item \isa {allI} (theorem), \bold{70}
  \item append function, 10--14
  \item \isacommand {apply} (command), 15
  \item \isa {arg_cong} (theorem), \bold{86}
  \item \isa {arith} (method), 23, 139
  \item arithmetic operations
    \subitem for \protect\isa{nat}, 23
  \item \textsc {ascii} symbols, \bold{195}
  \item associative-commutative function, 164
  \item \isa {assumption} (method), 59
  \item assumptions
    \subitem of subgoal, 12
    \subitem renaming, 72--73
    \subitem reusing, 73
  \item \isa {auto} (method), 37, 82
  \item \isa {axclass}, 152--158
  \item axiom of choice, 76
  \item axiomatic type classes, 152--158

  \indexspace

  \item \isacommand {back} (command), 68
  \item \isa {Ball} (constant), 99
  \item \isa {ballI} (theorem), \bold{98}
  \item \isa {best} (method), 82
  \item \isa {Bex} (constant), 99
  \item \isa {bexE} (theorem), \bold{98}
  \item \isa {bexI} (theorem), \bold{98}
  \item \isa {bij_def} (theorem), \bold{100}
  \item bijections, 100
  \item binary trees, 18
  \item binomial coefficients, 99
  \item bisimulations, 106
  \item \isa {blast} (method), 79--80, 82
  \item \isa {bool} (type), 4, 5
  \item boolean expressions example, 20--22
  \item \isa {bspec} (theorem), \bold{98}
  \item \isacommand{by} (command), 63

  \indexspace

  \item \isa {card} (constant), 99
  \item \isa {card_Pow} (theorem), \bold{99}
  \item \isa {card_Un_Int} (theorem), \bold{99}
  \item cardinality, 99
  \item \isa {case} (symbol), 32, 33
  \item \isa {case} expressions, 5, 6, 18
  \item case distinctions, 19
  \item case splits, \bold{31}
  \item \isa {case_tac} (method), 19, 91, 147
  \item \isa {clarify} (method), 81, 82
  \item \isa {clarsimp} (method), 81, 82
  \item \isa {classical} (theorem), \bold{63}
  \item coinduction, \bold{106}
  \item \isa {Collect} (constant), 99
  \item \isa {comp_def} (theorem), \bold{102}
  \item compiling expressions example, 36--38
  \item \isa {Compl_iff} (theorem), \bold{96}
  \item complement
    \subitem of a set, 95
  \item composition
    \subitem of functions, \bold{100}
    \subitem of relations, \bold{102}
  \item conclusion
    \subitem of subgoal, 12
  \item conditional expressions, \see{\isa{if} expressions}{1}
  \item conditional simplification rules, 31
  \item \isa {cong} (attribute), 164
  \item congruence rules, \bold{163}
  \item \isa {conjE} (theorem), \bold{61}
  \item \isa {conjI} (theorem), \bold{58}
  \item \isa {Cons} (constant), 9
  \item \isacommand {constdefs} (command), 25
  \item \isacommand {consts} (command), 10
  \item contrapositives, 63
  \item converse
    \subitem of a relation, \bold{102}
  \item \isa {converse_iff} (theorem), \bold{102}
  \item CTL, 111--116, 179--181

  \indexspace

  \item \isacommand {datatype} (command), 9, 38--43
  \item datatypes, 17--22
    \subitem and nested recursion, 40, 44
    \subitem mutually recursive, 38
    \subitem nested, 168
  \item \isacommand {defer} (command), 16, 90
  \item Definitional Approach, 26
  \item definitions, \bold{25}
    \subitem unfolding, \bold{30}
  \item \isacommand {defs} (command), 25
  \item \isa {del} (modifier), 29
  \item description operators, 75--77
  \item descriptions
    \subitem definite, 75
    \subitem indefinite, 76
  \item \isa {dest} (attribute), 92
  \item destruction rules, 61
  \item \isa {diff_mult_distrib} (theorem), \bold{141}
  \item difference
    \subitem of sets, \bold{96}
  \item \isa {disjCI} (theorem), \bold{64}
  \item \isa {disjE} (theorem), \bold{60}
  \item \isa {div} (symbol), 23
  \item divides relation, 74, 85, 91--94, 142
  \item division
    \subitem by negative numbers, 143
    \subitem by zero, 142
    \subitem for type \protect\isa{nat}, 141
  \item domain
    \subitem of a relation, 102
  \item \isa {Domain_iff} (theorem), \bold{102}
  \item \isacommand {done} (command), 13
  \item \isa {drule_tac} (method), 66, 86
  \item \isa {dvd_add} (theorem), \bold{142}
  \item \isa {dvd_anti_sym} (theorem), \bold{142}
  \item \isa {dvd_def} (theorem), \bold{142}

  \indexspace

  \item \isa {elim!} (attribute), 121
  \item elimination rules, 59--60
  \item \isacommand {end} (command), 14
  \item \isa {Eps} (constant), 99
  \item equality, 5
    \subitem of functions, \bold{99}
    \subitem of records, 151
    \subitem of sets, \bold{96}
  \item \isa {equalityE} (theorem), \bold{96}
  \item \isa {equalityI} (theorem), \bold{96}
  \item \isa {erule} (method), 60
  \item \isa {erule_tac} (method), 66
  \item Euclid's algorithm, 91--94
  \item even numbers
    \subitem defining inductively, 117--121
  \item \texttt {EX}, \bold{195}
  \item \isa {Ex} (constant), 99
  \item \isa {exE} (theorem), \bold{72}
  \item \isa {exI} (theorem), \bold{72}
  \item \isa {ext} (theorem), \bold{99}
  \item extensionality
    \subitem for functions, \bold{99, 100}
    \subitem for records, 151
    \subitem for sets, \bold{96}
  \item \ttEXU, \bold{195}

  \indexspace

  \item \isa {False} (constant), 5
  \item \isa {fast} (method), 82, 114
  \item Fibonacci function, 46
  \item \isa {finite} (symbol), 99
  \item \isa {Finites} (constant), 99
  \item fixed points, 106
  \item flags, 5, 6, 33
    \subitem setting and resetting, 5
  \item \isa {force} (method), 81, 82
  \item formulae, 5
  \item forward proof, 82--88
  \item \isa {frule} (method), 73
  \item \isa {frule_tac} (method), 66
  \item \isa {fst} (constant), 24
  \item function types, 5
  \item functions, 99--101
    \subitem partial, 170
    \subitem total, 11, 46--52
    \subitem underdefined, 171

  \indexspace

  \item \isa {gcd} (constant), 83--84, 91--94
  \item generalizing for induction, 119
  \item generalizing induction formulae, 34
  \item Girard, Jean-Yves, \fnote{61}
  \item Gordon, Mike, 3
  \item grammars
    \subitem defining inductively, 130--135
  \item ground terms example, 125--130

  \indexspace

  \item \isa {hd} (constant), 17, 37
  \item Hilbert's $\varepsilon$-operator, 76
  \item HOLCF, 43
  \item Hopcroft, J. E., 135
  \item \isa {hypreal} (type), 145

  \indexspace

  \item \isa {Id_def} (theorem), \bold{102}
  \item \isa {id_def} (theorem), \bold{100}
  \item identifiers, \bold{6}
    \subitem qualified, \bold{4}
  \item identity function, \bold{100}
  \item identity relation, \bold{102}
  \item \isa {if} expressions, 5, 6
    \subitem simplification of, 33
    \subitem splitting of, 31, 49
  \item if-and-only-if, 5
  \item \isa {iff} (attribute), 80, 92, 120
  \item \isa {iffD1} (theorem), \bold{84}
  \item \isa {iffD2} (theorem), \bold{84}
  \item image
    \subitem under a function, \bold{101}
    \subitem under a relation, \bold{102}
  \item \isa {image_def} (theorem), \bold{101}
  \item \isa {Image_iff} (theorem), \bold{102}
  \item \isa {impI} (theorem), \bold{62}
  \item implication, 62--63
  \item \isa {ind_cases} (method), 121
  \item \isa {induct_tac} (method), 12, 19, 52, 178
  \item induction, 174--181
    \subitem complete, 176
    \subitem deriving new schemas, 178
    \subitem on a term, 175
    \subitem recursion, 51--52
    \subitem structural, 19
    \subitem well-founded, 105
  \item induction heuristics, 33--35
  \item \isacommand {inductive} (command), 117
  \item inductive definition
    \subitem simultaneous, 131
  \item inductive definitions, 117--135
  \item \isacommand {inductive\_cases} (command), 121, 129
  \item infinitely branching trees, 42
  \item \isacommand{infixr} (annotation), 10
  \item \isa {inj_on_def} (theorem), \bold{100}
  \item injections, 100
  \item \isa {insert} (constant), 97
  \item \isa {insert} (method), 87--88
  \item instance, \bold{153}
  \item \texttt {INT}, \bold{195}
  \item \texttt {Int}, \bold{195}
  \item \isa {int} (type), 143
  \item \isa {INT_iff} (theorem), \bold{98}
  \item \isa {IntD1} (theorem), \bold{95}
  \item \isa {IntD2} (theorem), \bold{95}
  \item integers, 143
  \item \isa {INTER} (constant), 99
  \item \texttt {Inter}, \bold{195}
  \item \isa {Inter_iff} (theorem), \bold{98}
  \item intersection, 95
    \subitem indexed, 98
  \item \isa {IntI} (theorem), \bold{95}
  \item \isa {intro} (method), 64
  \item \isa {intro!} (attribute), 118
  \item \isa {intro_classes} (method), 153
  \item introduction rules, 58--59
  \item \isa {inv} (constant), 76
  \item \isa {inv_image_def} (theorem), \bold{105}
  \item inverse
    \subitem of a function, \bold{100}
    \subitem of a relation, \bold{102}
  \item inverse image
    \subitem of a function, 101
    \subitem of a relation, 104
  \item \isa {itrev} (constant), 34

  \indexspace

  \item \isacommand {kill} (command), 16

  \indexspace

  \item $\lambda$ expressions, 5
  \item LCF, 43
  \item \isa {LEAST} (symbol), 23, 75
  \item least number operator, \see{\protect\isa{LEAST}}{75}
  \item \isacommand {lemma} (command), 13
  \item \isacommand {lemmas} (command), 83, 92
  \item \isa {length} (symbol), 18
  \item \isa {length_induct}, \bold{178}
  \item \isa {less_than} (constant), 104
  \item \isa {less_than_iff} (theorem), \bold{104}
  \item \isa {let} expressions, 5, 6, 31
  \item \isa {Let_def} (theorem), 31
  \item \isa {lex_prod_def} (theorem), \bold{105}
  \item lexicographic product, \bold{105}, 166
  \item {\texttt{lfp}}
    \subitem applications of, \see{CTL}{106}
  \item linear arithmetic, 22--24, 139
  \item \isa {List} (theory), 17
  \item \isa {list} (type), 4, 9, 17
  \item \isa {list.split} (theorem), 32
  \item \isa {lists_mono} (theorem), \bold{127}
  \item Lowe, Gavin, 184--185

  \indexspace

  \item \isa {Main} (theory), 4
  \item major premise, \bold{65}
  \item \isa {max} (constant), 23, 24
  \item measure functions, 47, 104
  \item \isa {measure_def} (theorem), \bold{105}
  \item meta-logic, \bold{70}
  \item methods, \bold{16}
  \item \isa {min} (constant), 23, 24
  \item \isa {mod} (symbol), 23
  \item \isa {mod_div_equality} (theorem), \bold{141}
  \item \isa {mod_mult_distrib} (theorem), \bold{141}
  \item model checking example, 106--116
  \item \emph{modus ponens}, 57, 62
  \item \isa {mono_def} (theorem), \bold{106}
  \item monotone functions, \bold{106}, 129
    \subitem and inductive definitions, 127--128
  \item \isa {more} (constant), 148, 149
  \item \isa {mp} (theorem), \bold{62}
  \item \isa {mult_ac} (theorems), 142
  \item multiple inheritance, \bold{157}
  \item multiset ordering, \bold{105}

  \indexspace

  \item \isa {nat} (type), 4, 22, 141--142
  \item \isa {nat_less_induct} (theorem), 176
  \item natural deduction, 57--58
  \item natural numbers, 22, 141--142
  \item Needham-Schroeder protocol, 183--185
  \item negation, 63--65
  \item \isa {Nil} (constant), 9
  \item \isa {no_asm} (modifier), 29
  \item \isa {no_asm_simp} (modifier), 29
  \item \isa {no_asm_use} (modifier), 29
  \item non-standard reals, 145
  \item \isa {None} (constant), \bold{24}
  \item \isa {notE} (theorem), \bold{63}
  \item \isa {notI} (theorem), \bold{63}
  \item numbers, 139--145
  \item numeric literals, 140
    \subitem for type \protect\isa{nat}, 141
    \subitem for type \protect\isa{real}, 144

  \indexspace

  \item \isa {O} (symbol), 102
  \item \texttt {o}, \bold{195}
  \item \isa {o_def} (theorem), \bold{100}
  \item \isa {OF} (attribute), 85--86
  \item \isa {of} (attribute), 83, 86
  \item \isa {only} (modifier), 29
  \item \isacommand {oops} (command), 13
  \item \isa {option} (type), \bold{24}
  \item ordered rewriting, \bold{164}
  \item overloading, 23, 152--154
    \subitem and arithmetic, 140

  \indexspace

  \item pairs and tuples, 24, 145--148
  \item parent theories, \bold{4}
  \item pattern matching
    \subitem and \isacommand{recdef}, 47
  \item patterns
    \subitem higher-order, \bold{165}
  \item PDL, 108--110
  \item \isacommand {pr} (command), 16, 90
  \item \isacommand {prefer} (command), 16, 90
  \item primitive recursion, \see{recursion, primitive}{1}
  \item \isacommand {primrec} (command), 10, 18, 38--43
  \item product type, \see{pairs and tuples}{1}
  \item Proof General, \bold{7}
  \item proof state, 12
  \item proofs
    \subitem abandoning, \bold{13}
    \subitem examples of failing, 77--79
  \item protocols
    \subitem security, 183--193

  \indexspace

  \item quantifiers, 5
    \subitem and inductive definitions, 125--127
    \subitem existential, 72
    \subitem for sets, 98
    \subitem instantiating, 74
    \subitem universal, 69--72

  \indexspace

  \item \isa {r_into_rtrancl} (theorem), \bold{102}
  \item \isa {r_into_trancl} (theorem), \bold{103}
  \item range
    \subitem of a function, 101
    \subitem of a relation, 102
  \item \isa {range} (symbol), 101
  \item \isa {Range_iff} (theorem), \bold{102}
  \item \isa {Real} (theory), 145
  \item \isa {real} (type), 144--145
  \item real numbers, 144--145
  \item \isacommand {recdef} (command), 46--52, 104, 166--174
    \subitem and numeric literals, 140
  \item \isa {recdef_cong} (attribute), 170
  \item \isa {recdef_simp} (attribute), 48
  \item \isa {recdef_wf} (attribute), 168
  \item \isacommand {record} (command), 148
  \item \isa {record_split} (method), 151
  \item records, 148--152
    \subitem extensible, 149--150
  \item recursion
    \subitem guarded, 171
    \subitem primitive, 18
    \subitem well-founded, \bold{167}
  \item recursion induction, 51--52
  \item \isacommand {redo} (command), 16
  \item reflexive and transitive closure, 102--104
  \item reflexive transitive closure
    \subitem defining inductively, 122--125
  \item relations, 101--104
    \subitem well-founded, 104--105
  \item \isa {rename_tac} (method), 72--73
  \item \isa {rev} (constant), 10--14, 34
  \item rewrite rules, \bold{27}
    \subitem permutative, \bold{164}
  \item rewriting, \bold{27}
  \item \isa {rotate_tac} (method), 30
  \item \isa {rtrancl_refl} (theorem), \bold{102}
  \item \isa {rtrancl_trans} (theorem), \bold{102}
  \item rule induction, 118--120
  \item rule inversion, 120--121, 129--130
  \item \isa {rule_format} (attribute), 175
  \item \isa {rule_tac} (method), 66
    \subitem and renaming, 73

  \indexspace

  \item \isa {safe} (method), 81, 82
  \item safe rules, \bold{80}
  \item \isa {set} (type), 4, 95
  \item set comprehensions, 97--98
  \item \isa {set_ext} (theorem), \bold{96}
  \item sets, 95--99
    \subitem finite, 99
    \subitem notation for finite, \bold{97}
  \item settings, \see{flags}{1}
  \item \isa {show_brackets} (flag), 6
  \item \isa {show_types} (flag), 5, 16
  \item \isa {simp} (attribute), 11, 28
  \item \isa {simp} (method), \bold{28}
  \item \isa {simp} del (attribute), 28
  \item \isa {simp_all} (method), 28, 37
  \item simplification, 27--33, 163--166
    \subitem of \isa{let}-expressions, 31
    \subitem with definitions, 30
    \subitem with/of assumptions, 29
  \item simplification rule, 165--166
  \item simplification rules, 28
    \subitem adding and deleting, 29
  \item \isa {simplified} (attribute), 83, 86
  \item \isa {size} (constant), 17
  \item \isa {snd} (constant), 24
  \item \isa {SOME} (symbol), 76
  \item \texttt {SOME}, \bold{195}
  \item \isa {Some} (constant), \bold{24}
  \item \isa {some_equality} (theorem), \bold{76}
  \item \isa {someI} (theorem), \bold{76}
  \item \isa {someI2} (theorem), \bold{76}
  \item \isa {someI_ex} (theorem), \bold{77}
  \item sorts, 158
  \item \isa {spec} (theorem), \bold{70}
  \item \isa {split} (attribute), 32
  \item \isa {split} (constant), 145
  \item \isa {split} (method), 31, 146
  \item \isa {split} (modifier), 32
  \item split rule, \bold{32}
  \item \isa {split_if} (theorem), 32
  \item \isa {split_if_asm} (theorem), 32
  \item \isa {ssubst} (theorem), \bold{67}
  \item structural induction, \see{induction, structural}{1}
  \item subclasses, 152, 156
  \item subgoal numbering, 46
  \item \isa {subgoal_tac} (method), 88
  \item subgoals, 12
  \item subset relation, \bold{96}
  \item \isa {subsetD} (theorem), \bold{96}
  \item \isa {subsetI} (theorem), \bold{96}
  \item \isa {subst} (method), 67
  \item substitution, 67--69
  \item \isa {Suc} (constant), 22
  \item \isa {surj_def} (theorem), \bold{100}
  \item surjections, 100
  \item \isa {sym} (theorem), \bold{84}
  \item syntax, 6, 11
  \item syntax translations, 26

  \indexspace

  \item tacticals, 89
  \item tactics, 12
  \item \isacommand {term} (command), 16
  \item term rewriting, \bold{27}
  \item termination, \see{functions, total}{1}
  \item terms, 5
  \item \isa {THE} (symbol), 75
  \item \isa {the_equality} (theorem), \bold{75}
  \item \isa {THEN} (attribute), \bold{84}, 86, 92
  \item \isacommand {theorem} (command), \bold{11}, 13
  \item theories, 4
    \subitem abandoning, \bold{16}
  \item \isacommand {theory} (command), 16
  \item theory files, 4
  \item \isacommand {thm} (command), 16
  \item \isa {tl} (constant), 17
  \item \isa {ToyList} example, 9--15
  \item \isa {trace_simp} (flag), 33
  \item tracing the simplifier, \bold{33}
  \item \isa {trancl_trans} (theorem), \bold{103}
  \item transition systems, 107
  \item \isacommand {translations} (command), 26
  \item tries, 43--46
  \item \isa {True} (constant), 5
  \item tuples, \see{pairs and tuples}{1}
  \item \isacommand {typ} (command), 16
  \item type constraints, \bold{6}
  \item type constructors, 4
  \item type inference, \bold{5}
  \item type synonyms, 25
  \item type variables, 5
  \item \isacommand {typedecl} (command), 107, 158
  \item \isacommand {typedef} (command), 159--162
  \item types, 4--5
    \subitem declaring, 158--159
    \subitem defining, 159--162
  \item \isacommand {types} (command), 25

  \indexspace

  \item Ullman, J. D., 135
  \item \texttt {UN}, \bold{195}
  \item \texttt {Un}, \bold{195}
  \item \isa {UN_E} (theorem), \bold{98}
  \item \isa {UN_I} (theorem), \bold{98}
  \item \isa {UN_iff} (theorem), \bold{98}
  \item \isa {Un_subset_iff} (theorem), \bold{96}
  \item \isacommand {undo} (command), 16
  \item unification, 66--69
  \item \isa {UNION} (constant), 99
  \item \texttt {Union}, \bold{195}
  \item union
    \subitem indexed, 98
  \item \isa {Union_iff} (theorem), \bold{98}
  \item \isa {unit} (type), 24
  \item unknowns, 6, \bold{58}
  \item unsafe rules, \bold{80}
  \item updating a function, \bold{99}

  \indexspace

  \item variables, 6--7
    \subitem schematic, 6
    \subitem type, 5
  \item \isa {vimage_def} (theorem), \bold{101}

  \indexspace

  \item Wenzel, Markus, vii
  \item \isa {wf_induct} (theorem), \bold{105}
  \item \isa {wf_inv_image} (theorem), \bold{105}
  \item \isa {wf_less_than} (theorem), \bold{104}
  \item \isa {wf_lex_prod} (theorem), \bold{105}
  \item \isa {wf_measure} (theorem), \bold{105}
  \item \isa {wf_subset} (theorem), 168
  \item \isa {while} (constant), 173
  \item \isa {While_Combinator} (theory), 173
  \item \isa {while_rule} (theorem), 173

  \indexspace

  \item \isa {zadd_ac} (theorems), 143
  \item \isa {zmult_ac} (theorems), 143

\end{theindex}