Container (type theory)¶

A shape-and-positions representation of strictly positive collection-like type constructors in dependent type theory.

Version: v1 · 2026-09-08 · History
Domain-specific #: 3875
Origin domain: type theory
Subdomain: type theory

Core Idea¶

A unary container consists of a shape type S and a position family P indexed by S; its extension at element type X is the dependent sum of a shape and a function from positions to X. Structure is separated from contents: choose a shape, obtain its valid position type and label each position with an element, making maps and generic traversals operate uniformly across lists, trees and related functors. 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.

Scope of Application¶

Container (type theory) belongs to type theory and is useful where the analyst can specify the typed type theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the universe and type theory, shape type, dependent position family, extension formula, element type and labeling function, morphism or functor action, represented datatype and limits such as strict positivity are explicit. The scope is broad within that domain but bounded by the need for the universe and type theory, shape type, dependent position family, extension formula, element type and labeling function, morphism or functor action, represented datatype and limits such as strict positivity are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the universe and type theory, shape type, dependent position family, extension formula, element type and labeling function, morphism or functor action, represented datatype and limits such as strict positivity are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

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 Container (type theory). Container (type theory) 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: the typed type theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the universe and type theory, shape type, dependent position family, extension formula, element type and labeling function, morphism or functor action, represented datatype and limits such as strict positivity are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of type theory because they reuse the typed type theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Structure is separated from contents: choose a shape, obtain its valid position type and label each position with an element, making maps and generic traversals operate uniformly across lists, trees and related functors., and type the carrier, state every parameter and convention in the definition, test that the universe and type theory, shape type, dependent position family, extension formula, element type and labeling function, morphism or functor action, represented datatype and limits such as strict positivity are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Container (type theory) Domain-specific

Parents (1) — more general patterns this builds on

Container (type theory) is a kind of Abstract Data Type Prime

The proposed strict upward parent is prime:abstract_data_type.

Hierarchy paths (3) — routes to 2 parentless roots

Container (type theory) → Abstract Data Type → Information Hiding → Abstraction

Show alternative paths (2)

Neighborhood in Abstraction Space¶

Container (type theory) sits in a crowded region of the domain-specific corpus (5^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Mathematical Types, Functions & Infinity (33 abstractions)

Nearest neighbors

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