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22. Cubical Type Theory 본문
Cubical Type Theory
Cubical Type Theory is an extension of traditional type theory that introduces new features to better handle higher-dimensional structures and reasoning.
In Cubical Type Theory, the key innovation is the use of cubical sets, which provide a geometric interpretation of types and allow for the direct manipulation of paths and higher-dimensional structures.
Key features of Cubical Type Theory include:
- Cubical sets: Geometric objects used to model types and paths in the theory.
- Cubical operations: Operations that allow for the manipulation of paths and higher-dimensional structures, such as transport and composition.
- Cubical type constructors: Constructors that enable the creation and manipulation of higher-dimensional types, such as interval types and higher inductive types.
- Cubical type checking: Algorithms for checking the validity of cubical proofs and ensuring coherence of higher-dimensional structures.
Cubical Type Theory provides a powerful framework for formalizing and reasoning about mathematical and computational concepts involving higher-dimensional structures.
Lab Activity: Implementing Interval Types in Agda
Let's implement interval types in Agda, a dependently-typed programming language, to demonstrate Cubical Type Theory.
module Interval where
data I : Set where
zero one : I
seg : zero ≡ one
_∨_ : I → I → I
zero ∨ i = i
one ∨ _ = one
infixr 5 _∨_
seg_suc : ∀ (i : I) → (i ∨ i) ≡ i
seg_suc zero = refl
seg_suc one = refl
seg_suc seg = refl
In this lab activity, we define an interval type `I` in Agda, which represents the unit interval [0,1].
We then define the operation `_∨_`, which represents concatenation of intervals.
Finally, we define a proof `seg_suc` that shows the segment concatenated with itself is equal to the segment.
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