منابع مشابه
Canonicity for Cubical Type Theory
Cubical type theory is an extension of Martin-Löf type theory recently proposed by Cohen, Coquand, Mörtberg and the author which allows for direct manipulation of n-dimensional cubes and where Voevodsky’s Univalence Axiom is provable. In this paper we prove canonicity for cubical type theory: any natural number in a context build from only name variables is judgmentally equal to a numeral. To a...
متن کاملCartesian 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...
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ژورنال
عنوان ژورنال: Journal of Automated Reasoning
سال: 2018
ISSN: 0168-7433,1573-0670
DOI: 10.1007/s10817-018-9469-1