typed lambda-calculus - definition. What is typed lambda-calculus
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FORMAL SYSTEM IN MATHEMATICAL LOGIC
Simply-typed lambda calculus; Simply typed lambda; Bidirectional type checking; Simply typed lambda-calculus; Simply typed λ-calculus; Simply typed λ calculus; Λ→

typed lambda-calculus      
<theory> (TLC) A variety of lambda-calculus in which every term is labelled with a type. A function application (A B) is only synctactically valid if A has type s --> t, where the type of B is s (or an instance or s in a polymorphic language) and t is any type. If the types allowed for terms are restricted, e.g. to Hindley-Milner types then no term may be applied to itself, thus avoiding one kind of non-terminating evaluation. Most functional programming languages, e.g. Haskell, ML, are closely based on variants of the typed lambda-calculus. (1995-03-25)
Typed lambda calculus         
TYPED FORMALISM THAT USES THE LAMBDA-SYMBOL (Λ) TO DENOTE ANONYMOUS FUNCTION ABSTRACTION
Typed lambda calculi
A typed lambda calculus is a typed formalism that uses the lambda-symbol (\lambda) to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered (see kinds below).
beta conversion         
<theory> A term from lambda-calculus for beta reduction or beta abstraction. (1999-01-15)

ويكيبيديا

Simply typed lambda calculus

The simply typed lambda calculus ( λ {\displaystyle \lambda ^{\to }} ), a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor ( {\displaystyle \to } ) that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical use of the untyped lambda calculus.

The term simple type is also used to refer extensions of the simply typed lambda calculus such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). In contrast, systems which introduce polymorphic types (like System F) or dependent types (like the Logical Framework) are not considered simply typed. The simple types, except for full recursion, are still considered simple because the Church encodings of such structures can be done using only {\displaystyle \to } and suitable type variables, while polymorphism and dependency cannot.