Programming in Martin-Löf's Type Theory

data type, 179 abstraction, 14 Abstraction rule, 143 absurdity, 43 append, 68 application, 13 Application rule, 143 apply, 48, 148 arity of an expression, 18 Assumption rule, 123, 142 AUTOMATH, 8 axiom of choice, 115 Bishop, 6 Bool, 44 Bool rules, 44–45 Boolean, 44–45 Calculus of Constructions, 8 canonical expression, 26 cartesian product of a family of sets, 47–55, 148 case{i1,...,in}, 41 case, 151 Church, 17 combination, 15 combined expression, 17 conjunction, 74 cons, 67, 152 Constable, 6, 167 constructive mathematics, 6 context, 29, 139 Coquand, 3, 7, 8 Curry, 10 Curry-Howard interpretation, 6 Cut rule for elements in sets, 124 Cut rule for equal elements in sets, 124 Cut rule for equal sets, 123 Cut rule for propositions, 123 Cut rule for true propositions, 124 de Bruijn, 8, 10 definiendum, 19 definiens, 19 definitional equality of expressions, 15 Dijkstra, 167, 171 disjoint union of a family of sets, 81– 82, 149 disjoint union of two sets, 79–80, 150 Dummet, 6 El formation rule, 139 elimination rule, 35 empty set, 43 enumeration sets, 41–42, 151 Eq, 57 Eq rules, 60–61 equality as a proposition, 57, 117 between canonical elements, 27 between elements in a set, 29, 31, 37, 119, 121 between expressions, 15 between sets, 28, 30, 37, 118, 121 extensional, 57 identity between objects in a type, 138 identity between types, 138 intensional, 57 equality rule, 35 equality sets, 57–62, 150 examples: Bool has only the elements true and false, 161 A partitioning problem, 171 Decidable predicates, 162 Division by 2, 155 Even or odd, 159 Module for sets with a computable equality, 182 Stack of A elements, 182 Stack of natural numbers, 179 Stronger elimination rules, 163 existential quantifier, 81, 82 expressions, 13 arity, 18 canonical —, 26 combined —, 17 saturated —, 17 single —, 17 unsaturated —, 17 extensional equality, 57 wfalse, 44 Fermat’s last theorem, 4 formation rule, 35 Frege, 17