Implementation of Two Layers Type Theory in Dedukti and Application to Cubical Type Theory
نویسندگان
چکیده
In this paper, we make a substantial step towards an encoding of Cubical Type Theory (CTT) in the Dedukti logical framework. Type-checking CTT expressions features decision procedure de Morgan algebra that so far could not be expressed by rewrite rules Dedukti. As alternative, 2 Layer Theories are variants Martin-L\"of where all or part definitional equality can represented terms so-called external equality. We propose to split giving (2LTT) Dedukti, and partial 2LTT.
منابع مشابه
Cartesian Cubical Type Theory
We present a cubical type theory based on the Cartesian cube category (faces, degeneracies, symmetries, diagonals, but no connections or reversal) with univalent universes, each containing Π, Σ, path, identity, natural number, boolean, pushout, and glue (equivalence extension) types. The type theory includes a syntactic description of a uniform Kan operation, along with judgemental equality rul...
متن کاملCubical Type Theory
() : ∆→ () σ : ∆→ Γ ∆ ` u : Aσ (σ, x = u) : ∆→ Γ, x : A σ : ∆→ Γ ∆ ` φ : I (σ, i = φ) : ∆→ Γ, i : I σ : ∆→ Γ Γ ` A ∆ ` Aσ σ : ∆→ Γ Γ ` t : A ∆ ` tσ : Aσ We can define 1Γ : Γ→ Γ by induction on Γ and then if Γ ` u : A we write (x = u) : Γ→ Γ, x : A for 1Γ, x = u. If we have further Γ, x : A ` t : B we may write t(u) and B(u) respectively instead of t(x = u) and B(x = u). Similarly if Γ ` φ : I w...
متن کاملCubical Type Theory
The equality on the inverval I is the equality in the free bounded distributive lattice on generators i, 1− i. The equality in the face lattice F is the one for the free distributive lattice on formal generators (i = 0), (i = 1) with the relation (i = 0) ∧ (i = 1) = 0. We have [(r ∨ s) = 1] = (r = 1) ∨ (s = 1) and [(r∧s) = 1] = (r = 1)∧ (s = 1). An irreducible element of this lattice is a face,...
متن کاملGuarded Cubical Type Theory
Guarded dependent type theory [1] is a dependent type theory with guarded recursive types, which are useful for building models of program logics, and as a tool for programming and reasoning with coinductive types. This is done via a modality ., pronounced ‘later’, with a constructor next, and a guarded fixed-point combinator fix : (.A → A) → A. This combinator is used both to define guarded re...
متن کاملCartesian Cubical Computational Type Theory
We present a dependent type theory organized around a Cartesian notion of cubes (with faces, degeneracies, and diagonals), supporting both fibrant and non-fibrant types. The fibrant fragment includes Voevodsky’s univalence axiom and a circle type, while the non-fibrant fragment includes exact (strict) equality types satisfying equality reflection. Our type theory is defined by a semantics in cu...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Electronic proceedings in theoretical computer science
سال: 2021
ISSN: ['2075-2180']
DOI: https://doi.org/10.4204/eptcs.332.4