Typing rule¶

A formal inference rule specifying how types of component expressions and contextual assumptions justify a type judgment for a larger syntactic construction.

Version: v1 · 2026-09-08 · History
Domain-specific #: 7307
Origin domain: type theory
Subdomain: typing judgments

Core Idea¶

A typing rule is an inference schema defining when a term has a type under a context. Rules recursively combine derivations for subterms, enforce constructor-specific constraints and produce a derivation whose root certifies the whole expression. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of type theory. It is local proof rule constructing global well-typedness derivations. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that premises, binding scope, context changes and conclusion follow the declared type system and substitution discipline fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test.

Scope of Application¶

Typing rule belongs to type theory and is useful where the analyst can specify a typing context Gamma, term syntax, types, premise judgments, conclusion judgment, rule schema, substitution and derivation tree, then evaluate premises, binding scope, context changes and conclusion follow the declared type system and substitution discipline. The scope is broad within that domain but bounded by the need for premises, binding scope, context changes and conclusion follow the declared type system and substitution discipline. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making premises, binding scope, context changes and conclusion follow the declared type system and substitution discipline the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Typing rule can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Typing rule. Typing rule compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: a typing context Gamma, term syntax, types, premise judgments, conclusion judgment, rule schema, substitution and derivation tree. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express premises, binding scope, context changes and conclusion follow the declared type system and substitution discipline independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of type theory because they reuse a typing context Gamma, term syntax, types, premise judgments, conclusion judgment, rule schema, substitution and derivation tree, Rules recursively combine derivations for subterms, enforce constructor-specific constraints and produce a derivation whose root certifies the whole expression., and type the carrier, state every parameter and convention in the definition, test that premises, binding scope, context changes and conclusion follow the declared type system and substitution discipline, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Typing rule Domain-specific

Parents (1) — more general patterns this builds on

Typing rule is a kind of Constraint Prime

The proposed strict upward parent is prime:constraint.

Hierarchy path (1) — routes to 1 parentless root

Typing rule → Constraint

Neighborhood in Abstraction Space¶

Typing rule sits in a crowded region of the domain-specific corpus (16^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Formal Logic & Type Theory (34 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08