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	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	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
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
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

based on cubical type theory:

cubicaltt
redtt
yacctt
RedPRL
Cubical Agda
- library and demos
- 1lab (cross-linked reference resource)

based on modal type theory:

Agda-flat: installation instructions

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 and Agda)
Xena project (using Lean)

Other proof assistants

Andromeda

Historical projects that died out:

The QED Project

References

Gentle exposition in

Théo Winterhalter, §1 of: Formalisation and Meta-Theory of Type Theory, Nantes (2020) [pdf, github]

Further exposition:

Jeremy Avigad, Interactive Theorem Proving, Automated Reasoning, and Mathematical Computation, (2012) pdf slides

On computer assisted proofs in analysis:

Oscar E. Lanford, Computer assisted proofs, in: Computational Methods in Field Theory, Lecture Notes in Physics, 409, Springer (1992) [doi:10.1007/3-540-55997-3_30]