nLab
proof assistant

Contents

Context

Constructivism, Realizability, Computability

Type theory

natural deduction metalanguage, practical foundations

judgement
hypothetical judgement, sequent
- antecedents $\vdash$ consequent, succedents

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

calculus of constructions

syntax object language

theory, axiom
proposition/type (propositions as types)
definition/proof/program (proofs as programs)
theorem

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

logic	category theory	type theory
true	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	initial object	empty type
proposition	(-1)-truncated object	h-proposition, mere proposition
proof	generalized element	program
cut rule	composition of classifying morphisms / pullback of display maps	substitution
cut elimination for implication	counit for hom-tensor adjunction	beta reduction
introduction rule for implication	unit for hom-tensor adjunction	eta conversion
logical conjunction	product	product type
disjunction	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	internal hom	function type
negation	internal hom into initial object	function type into empty type
universal quantification	dependent product	dependent product type
existential quantification	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
equivalence	path space object	identity type/path type
equivalence class	quotient	quotient type
induction	colimit	inductive type, W-type, M-type
higher induction	higher colimit	higher inductive type
-	0-truncated higher colimit	quotient inductive type
coinduction	limit	coinductive type
preset		type without identity types
completely presented set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	internal 0-groupoid	Bishop set/setoid
universe	object classifier	type of types
modality	closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic	(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net	string diagram	quantum circuit
(absence of) contraction rule	(absence of) diagonal	no-cloning theorem
	synthetic mathematics	domain specific embedded programming language

homotopy levels

semantics

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Idea
Examples
References

Idea

A proof assistant or proof management system is a kind of software designed to help with proofs in formalized mathematics. Many proof assistants resemble and/or include a programming language.

There are arguably two threads of current development in proof systems, which may be called “foundational” and “coverage”.

The “foundational” work tries to find the best foundational theory to formalize mathematics (see also at foundations of mathematics). Out of that work first came dependent types (Automath, in the late 60s), then the calculus of constructions (early Coq), and the calculus of inductive constructions (current Coq). More recently a new wave of such work is being done in homotopy type theory as another step in this direction. Coq’s library is not that large, except in the area of group theory where the results of the work on Feit-Thompson theorem has produced something larger.

The “coverage” work tries to formalize as much as possible of mathematics in existing theories. For instance, for decades people have been building Mizar‘s library (Mizar is based on Tarski–Grothendieck set theory rather than type theory). Its library is a couple of orders of magnitude larger than anyone else’s. On the other hand, despite this quantity, it remains an issue to attack problems of contemporary research interest in these systems, see also at Mizar – problem of pertinence.

Similar to Mizar is NuPRL, HOL light and Isabelle, which all have decently sized libraries. (Isabelle can be used with either material set theory, like Mizar, or higher-order type theory, like the others.)

Examples

proof assistants:

based on plain type theory/set theory:

Metamath
Mizar, NuPRL, Isabelle, HOL

based on dependent type theory/homotopy type theory:

Coq, Agda, Lean, Arend

For monoidal category theory:

DisCoPy

For higher category theory:

projects for formalization of mathematics with proof assistants:

Archive of Formal Proofs (using Isabelle)
ForMath project (using Coq)
MathScheme
UniMath project (using Coq)
Xena project (using Lean)

Historical projects that died out:

The QED Project

References

Wikipedia, Proof assistant
Oscar Lanford III., Computer assisted proofs in analysis, Proceedings of the International Congress of Mathematicians, 1986 (pdf)
Freek Wiedijk, Digital math by alphabet (web)
Carlos Simpson, Verification and creation of proofs by computer
Jeremy Avigad, Interactive Theorem Proving,

Automated Reasoning, and Mathematical Computation_, 2012 pdf slides
Conference Series on Intelligent Computer Mathematics

(2014)
Conference series on Interactive theorem proving

(2014)

Parts of the above text are taken from this MO comment by Jacques Carette.

Proof assistants specifically for homotopy type theory:

HoTT wiki, Proof Assistants

nLab proof assistant