Let expressionIn computer science, a "let" expression associates a function definition with a restricted scope. The "let" expression may also be defined in mathematics, where it associates a Boolean condition with a restricted scope. The "let" expression may be considered as a lambda abstraction applied to a value. Within mathematics, a let expression may also be considered as a conjunction of expressions, within an existential quantifier which restricts the scope of the variable.
Lambda liftingLambda lifting is a meta-process that restructures a computer program so that functions are defined independently of each other in a global scope. An individual "lift" transforms a local function into a global function. It is a two step process, consisting of; Eliminating free variables in the function by adding parameters. Moving functions from a restricted scope to broader or global scope. The term "lambda lifting" was first introduced by Thomas Johnsson around 1982 and was historically considered as a mechanism for implementing functional programming languages.
Fixed-point combinatorIn mathematics and computer science in general, a fixed point of a function is a value that is mapped to itself by the function. In combinatory logic for computer science, a fixed-point combinator (or fixpoint combinator) is a higher-order function that returns some fixed point of its argument function, if one exists. Formally, if the function f has one or more fixed points, then and hence, by repeated application, In the classical untyped lambda calculus, every function has a fixed point.
CoqCoq is an interactive theorem prover first released in 1989. It allows for expressing mathematical assertions, mechanically checks proofs of these assertions, helps find formal proofs, and extracts a certified program from the constructive proof of its formal specification. Coq works within the theory of the calculus of inductive constructions, a derivative of the calculus of constructions. Coq is not an automated theorem prover but includes automatic theorem proving tactics (procedures) and various decision procedures.
Decidability of first-order theories of the real numbersIn mathematical logic, a first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressions over real variables. The corresponding first-order theory is the set of sentences that are actually true of the real numbers. There are several different such theories, with different expressive power, depending on the primitive operations that are allowed to be used in the expression.
Affine logicAffine logic is a substructural logic whose proof theory rejects the structural rule of contraction. It can also be characterized as linear logic with weakening. The name "affine logic" is associated with linear logic, to which it differs by allowing the weakening rule. Jean-Yves Girard introduced the name as part of the geometry of interaction semantics of linear logic, which characterizes linear logic in terms of linear algebra; here he alludes to affine transformations on vector spaces. Affine logic predated linear logic.
Agda (programming language)Agda is a dependently typed functional programming language originally developed by Ulf Norell at Chalmers University of Technology with implementation described in his PhD thesis. The original Agda system was developed at Chalmers by Catarina Coquand in 1999. The current version, originally known as Agda 2, is a full rewrite, which should be considered a new language that shares a name and tradition. Agda is also a proof assistant based on the propositions-as-types paradigm, but unlike Coq, has no separate tactics language, and proofs are written in a functional programming style.
Map (higher-order function)In many programming languages, map is the name of a higher-order function that applies a given function to each element of a collection, e.g. a list or set, returning the results in a collection of the same type. It is often called apply-to-all when considered in functional form. The concept of a map is not limited to lists: it works for sequential containers, tree-like containers, or even abstract containers such as futures and promises. Suppose we have a list of integers [1, 2, 3, 4, 5] and would like to calculate the square of each integer.
Linear logicLinear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields such as programming languages, game semantics, and quantum physics (because linear logic can be seen as the logic of quantum information theory), as well as linguistics, particularly because of its emphasis on resource-boundedness, duality, and interaction.
