nLab
program

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-level 1-type/h-prop
proofgeneralized elementprogram
conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator monadmodal type theory, monad (in computer science)

homotopy levels

semantics

Contents

In type theory

In computer science a program is a construction of a term of a certain type (the output of the program).

Under computational trinitarianism this is essentially the same as a proof in type theory of the proposition presented by the type (see at propositions as types and proofs as programs).

Revised on September 6, 2012 19:11:11 by Urs Schreiber (131.174.188.45)
Edit | Back in time (1 revision) | See changes | History | Views: Print | TeX | Source | Linked from: foundation of mathematics, function, logic, inhabited set, internal logic, context, homotopy n-type, homotopy type, type theory, sequent calculus, computer science, type, preset, propositions as types, syntax - semantics duality, proposition, theory, dependent type, W-type, type theory - contents, first-order hyperdoctrine, tripos, identity type, calculus of constructions, algebraic model for modal logics, pure type system, the logic S4(m), modal logic, normal modal logic, term, variable, cartesian logic, hyperdoctrine, categorical semantics, syntactic category, cartesian theory, axiom, homotopy type theory, type of propositions, theorem, unit type, type of types, types and logic - table, axiom UIP, contents of contents, syntax, product type, quotient type, cohesive homotopy type theory, homotopy level, function type, bracket type, dependent product type, Coq, empty type, inductive type, Agda, higher inductive type, existential quantifier, universal quantifier, proofs as programs, proof, HoTT methods for homotopy theorists, univalence axiom, intensional type theory, judgmental equality, dependent type theory, contractible type, extensional type theory, sum type, semantics, Robert Harper, h-set, decidability, h-proposition, equivalence in homotopy type theory, canonical form, polymorphism, coherent hyperdoctrine, type formation, dependent sum type, substitution, fixed-point combinator, relation between type theory and category theory, M-type, categorical model of dependent types, sequent, first-order theory, Haskell, logical framework, Burali-Forti's paradox, judgment, Russell's paradox, context extension, numeral, definition, program, elementary function arithmetic, Brouwer-Heyting-Kolmogorov interpretation, second-order arithmetic, computational trinitarianism, monad (in computer science), computation, programming language, natural deduction, metalanguage, theory of presheaf type, Practical Foundations for Programming Languages, term in context, contradiction, inconsistency, modal type, type of h-sets, Bishop set, antecedent, succedent, consequent, looping combinator, deductive system, h-groupoid, coinductive type, practical foundation of mathematics, modal type theory, differential homotopy type theory, axiom K, observational type theory, meaning explanation, natural numbers type