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Beta equivalence is the smallest equivalence relation containing beta reduction; equivalently, $M{\equiv }_{\beta }N$ when a finite sequence of beta reductions and reversed beta reductions connects $M$ and $N$.

A beta-redex is a subterm of the form $(\lambda x.M)N$, to which beta reduction can be applied.

The omega combinator is the divergent untyped lambda calculus termwhose only beta reduction reproduces $\Omega$.

A partial function $f:{\mathbb{N}}^k\rightharpoonup \mathbb{N}$ is lambda-definable when a lambda term $F$ sends the corresponding Church numerals to the numeral for $f(\mathbf{n})$ whenever the value is defined and produces no numeral otherwise; in other words, it is a lambda-definable partial function. The initial functions are represented byand the appropriate variable in $\lambda x_1\dots x_k.x_i$. If $G,H_1,\dots ,H_m$ represent $g,h_1,\dots ,h_m$, thenrepresents their composition.

Church Booleans supply a lazy conditional, andtests whether a Church numeral is zero. A standard predecessor term isLet $G$ and $H$ represent the base and step functions of a primitive recursion. With a fixed-point combinator $Y$, defineNormal-order beta reduction evaluates only the selected branch. Induction on the input numeral gives the two recursion equations, so this is the lambda definition of primitive recursion.

For unbounded minimization, let $G$ represent $g(\mathbf{x},n)$ and defineThen $Mx0$ tests $0,1,2,\dots$ in order and returns the least zero of $g$. If no zero is reached, or a required earlier computation is undefined, reduction never produces a Church numeral. This is the lambda definition of unbounded minimization. Since the partial recursive functions are generated from the initial functions by composition, primitive recursion, and minimization, every partial computable function is represented by a lambda term on Church numerals.

Under the Curry-Howard correspondence, Intuitionistic propositional logic propositions are types and proofs are typed terms. Assumptions correspond to typed variables. The logical implication $A\to B$ corresponds to the function type, logical conjunction $A\wedge B$ to the product type, logical disjunction $A\vee B$ to the sum type, logical truth to the unit type, and logical falsity to the empty type.

The rules of natural deduction become term constructors. Implication introduction sends a derivation of $B$ from a variable $x:A$ to the abstraction $\lambda x.M:A\to B$, while implication elimination becomes application $MN$. Pairing and projection implement conjunction, and injections with case analysis implement disjunction. Under this correspondence, normalization of proofs is computation by beta reduction in the simply typed lambda calculus.

The Church numeral corresponding to the natural number $n$ isA function $g:{\mathbb{N}}^k\to \mathbb{N}$ is a lambda-definable function if some closed lambda term $G$ satisfiesfor all natural numbers $n_1,\dots ,n_k$.

DefineThen beta reduction givesTherefore the successor function is lambda-definable; this is the lambda definition of the successor function.

By assumption the set of combinators is recursively enumerable. Finite beta reduction sequences, and hence finite certificates of beta equivalence, are also recursively enumerable.

For a closed term $Y$, choose a fresh variable $f$. The term $Y$ is a fixed-point combinator exactly whenIndeed, substitution then gives the required equivalence for every $F$, and the forward direction follows by taking $F=f$. Dovetail the enumeration of closed terms with all finite beta-equivalence certificates. Whenever a certificate of the displayed equivalence is found, output $Y$. This enumerates exactly the fixed-point combinators, proving recursively enumerable fixed-point combinators.

A term is in beta-normal form when it contains no beta-redex, meaning no subterm of the formEquivalently, no beta reduction can be performed anywhere in the term.

The Weak normalization theorem for simply typed lambda calculus states that every well-typed term of the simply typed lambda calculus admits at least one finite sequence of beta reductions ending in a beta-normal form.