nLab MathScheme

Contents

Context

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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Foundations

Constructivism, Realizability, Computability

Idea
Related entries
References

Idea

MathScheme is a project aimed at combining computation via computer algebra? with formal proof via proof assistants.

MathScheme Web page

From their Mission page:

Sound mathematics is essential in all modern technological developments. As we build increasingly complex systems, mathematical reasoning becomes more difficult and error-prone. This is particularly true for software systems. A mechanized mathematics system (MMS) is required that can support, improve, and — where possible — automate mathematical reasoning.

There are presently two major types of MMSs, computer algebra systems and computer theorem proving systems. Computer algebra systems provide algorithms for symbolic computation. They are relatively fast and easy to use, but they are not rigorously based, untrustworthy, and narrow in scope. Computer theorem proving systems provide tools for creating formal proofs. They are based on well-defined logical foundations, can support a wide range of mathematics, but are difficult to use and often lack the specific knowledge needed to perform many routine computations.

While mechanized mathematics has the potential to revolutionize the design, implementation, and analysis of software systems, contemporary MMSs fall well short of this potential. There are no MMSs today that provide highly integrated symbolic computation and formal deduction capabilities. There have been earlier attempts to add the capabilities of one type of MMS to an existing system of the other type or to combine two existing systems of different type, but these attempts have not been very successful. One cannot add fundamental capabilities as an afterthought.

Our objective is to develop a new approach to mechanized mathematics that combines the strengths of symbolic algebra systems with those of formal deduction systems to yield a system that is both powerful and sound. The first goal of the project is to develop a formal framework that integrates symbolic computation and formal deduction. The framework should allow mathematical knowledge to be represented both declaratively using axioms and procedurally using algorithms, and it should provide a style of mathematical reasoning in which computation and deduction are intertwined.

The second project goal is to design and implement an MMS based on the formal framework. It should provide services for building formal languages, theories, computations, deductions, and mappings between theories — the services that are necessary for mechanizing the mathematics process.

Our long-range goal is to build, on top of the MMS, an interactive mathematics laboratory (IML) that provides an integrated set of tools for facilitating and managing mathematical reasoning. The IML is intended to have the capabilities of both contemporary computer algebra systems and computer theorem proving systems and the means to formalize a wide range of mathematical knowledge.