Typepreserving

Typepreserving refers to a property of transformations or translations in typed languages that preserve the types of expressions. In this sense, a transformation is typepreserving if, whenever an expression e has type τ under some typing context, the transformed expression e' has the same type τ (or an equivalent type) under the corresponding context. The notion is important in programming language theory, compiler design, and formal methods, where maintaining type information across steps of evaluation or translation helps ensure safety and correctness.

In formal language theory, the preservation of types is often formalized as a subject reduction (or type preservation) property: if e has type τ and e reduces to e', then e' has type τ. When combined with progress, this yields type safety: a well-typed program never goes wrong during evaluation.

In practice, typepreserving transformations are used by compilers and toolchains. Optimizations and translations should not introduce type errors. Examples include beta-reduction or alpha-conversion in a simply typed lambda calculus, constant folding, or other syntax-directed rewrites that preserve typing. In source-to-source transformations and intermediate representations, a type-preserving pass ensures the resulting code remains well-typed.

In languages with subtyping, preserving types may require more elaborate rules; a transformation might preserve a related type (for example, a supertype or an equivalent typing relation) or rely on subtyping to maintain safety. The property is central to proofs of compiler correctness and to reasoning about program equivalence in typed settings.

See also: type safety, subject reduction, typing, compiler correctness, type systems.