certified programming


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-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination? for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
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, (idemponent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels




A certification of a computer program is a formalized guarantee – a proof – that the program has given specified properties. For instance, it could be guaranteed to compute a given output based on a given input, or to always terminate, or to not include a certain kind of security hole.

Certifications often take the form of a proof that a program, regarded as a term of some sort (under programs as proofs), has a specified type. Thus, programming languages based on highly expressive type theories? (including dependent types) are a natural place to do certified programming “natively”. Examples are Coq and Agda. In this case, the program is written at the same time as a proof of its certification. One often then wants to “extract” the executable code or “ignore” the proof part of the terms when actually running the code, for performance reasons; Coq and Agda include mechanisms designed for this.

It is also possible to write a program in some less strongly typed language and provide an “external” certification for it, rather than one built into the program itself. Computer proof assistants like Coq and Agda are also used for this, using a formal representation of some other programming language. There are also other program analysis tools which can produce automated proofs of certain aspects of a computer program, such as safety and termination (although of course a complete solution to termination-checking is impossible, being the halting problem).

So far, fully certified programming in the type-theoretic sense is largely an academic endeavor, see for instance (SpittersKrebbersvdWeegen); the tools available at present usually require too much time and effort to be worth the payoff in industry. As automation progresses, this may change.



  • Adam Chlipala, Certified programming with dependent types (web)

  • Adam Chlipala, Implementing Certified Programming Language Tools in Dependent Type Theory PhD (2007) (web)

Discussion of the need for certified programming in scientific computation is in

  • Bas Spitters, Robberts Krebbers, Eelis van der Weegen, From computational analysis to thoughts about analysis in HoTT, MAP International spring school on formalization of Mathematics (2012) (pdf)
Revised on February 11, 2014 13:50:52 by Urs Schreiber (