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Results tagged with type-theory
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user 83968
formal systems to specify properties of objects
12
votes
2
answers
3k
views
Why does Coq include let-expressions in its core language
Coq includes let-expressions in its core language.
We can translate let-expressions to applications like this:
let x : t = v in b ~> (\(x:t). b) v
I understand that this does not always work because t …
- 575
5
votes
1
answer
104
views
What untyped term inhabits induction on natural numbers in CoC?
Induction on Church-encoded natural numbers (which I will call indNat) can not be defined within the Calculus of Constructions.
If we assumed indNat as an axiom, is there an untyped term that would h …
- 575
3
votes
0
answers
117
views
Calculus of constructions, type-in-type and recursion
Does adding type-in-type to the calculus of constructions lead to (general) recursion? Such that one can write the Y combinator.
- 575
3
votes
How does the `Word` type work in Kind Lang?
Giving the datatype in Agda-style syntax:
data Word : Nat -> Set where
e : Word zero
o : {size : Nat} -> Word size -> Word (suc size)
i : {size : Nat} -> Word size -> Word (suc size)
So e is an …
- 575
2
votes
1
answer
75
views
Restrictions needed on ADT for totality
In the paper Total Functional Programming by D.A. Turner three rules are given for a programming language to remain total:
complete case analysis
covariant type recursion (type constructor should no …
- 575
2
votes
0
answers
163
views
Is Observational Equality better than intensional equality?
The Observational Equality from Epigram 2 seems to be intensional equality (like Coq and Agda have), but it also supports function extensionality. In that sense it seems that Observational Equality is …
- 575
1
vote
0
answers
35
views
Is there a most general fixpoint?
We can write inductive types in terms of a fixpoint type:
Fix : (* -> *) -> *
In : (f : * -> *) -> f (Fix f) -> Fix f
NatF r = () + r
Nat = Fix NatF
Z = In NatF (InL ())
S n = In NatF (InR n)
But th …
- 575
1
vote
0
answers
45
views
Does Quantitative Type Theory make the Prop universe obsolete?
Coq (and other type theories such as Setoid Type Theory) have a Prop universe for propositions. As far as I understand this universe is needed to be sure that the propositions can be erased. In Quanti …
- 575