Guarded Cubical Type Theory
نویسندگان
چکیده
منابع مشابه
Guarded 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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This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type checking, while still supporting non-...
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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,...
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ژورنال
عنوان ژورنال: Journal of Automated Reasoning
سال: 2018
ISSN: 0168-7433,1573-0670
DOI: 10.1007/s10817-018-9471-7