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[Thorsten Altenkirch <Thorsten.Altenkirch AT nottingham.ac.uk>]

Hi,

Just a comment: using functional extensionality we can actually prove UIP
for types with a stable equality, that is ~~(x = y) -> x=y. Clearly we get
a constant function by composing with x = y -> ~~(x =y). Function types
(and Pi Types) are closed under stable equality, that is if B has a stable
equality then so has A -> B. Using this we can show that all the ³normal²
types in the 1st universe have UIP, the first proper counterexample is the
first universe itself if we assume univalence and higher inductive types.

Cheers,
Thorsten

On 15/12/2016, 09:05, 
"coq-club-request AT inria.fr
 on behalf of Chunhui He"
<coq-club-request AT inria.fr
 on behalf of 
iuhnuhceh AT gmail.com>
 wrote:

>Hi all,
>
>I'm reading the proof of UIP on decidable domains in the standard
>library of coq:
>source: theories/Logic/Eqdep_dec.v
>doc: https://coq.inria.fr/library/Coq.Logic.Eqdep_dec.html
>
>The proof "eq_proofs_unicity_on" uses some definitions like "nu" and
>"nu_inv".
>I think the proof is confusing, does anyone know the meaning of "nu"
>and "nu_inv"?
>
>I try to simplify the proof by inlining (see below).
>I find we can use a projection instead of "nu" and "nu_inv".
>maybe it seem cleaner than the original proof?
>
>Thanks!
>
>Chunhui
>
>---
>Section eqdep_dec.
>  Variable A : Type.
>  Variable A_dec : forall (x y : A), x = y \/ x <> y.
>
>  Definition or_proj {x' y : A} (oreq : x' = y \/ x' <> y)
>             (x : A) (def : x = y) : x = y :=
>    match oreq with
>    | or_introl H =>
>      match A_dec x' x with
>      | or_introl e =>
>        match e in _ = y' return y' = y with
>        | eq_refl => H
>        end
>      | or_intror _ => def
>      end
>    | or_intror _ => def
>    end.
>
>  Lemma or_proj_eq : forall {x y : A} (u : x = y), or_proj (A_dec x y) x
>u = u.
>  Proof.
>    intros x y u.
>    unfold or_proj.
>    case u.
>    case (A_dec x x) as [Heq | Hneq].
>    { case Heq. apply eq_refl. }
>    { exfalso; apply (Hneq eq_refl). }
>  Qed.
>
>  Theorem eq_proofs_unicity : forall {x y : A} (p1 p2 : x = y), p1 = p2.
>  Proof.
>    intros x y p1 p2.
>    case (or_proj_eq p1).
>    case (or_proj_eq p2).
>    unfold or_proj.
>    case (A_dec x y) as [Heq | Hneq].
>    { case (A_dec x x) as [Heq' | Hneq'] .
>      { apply eq_refl. }
>      { exfalso; apply (Hneq' eq_refl). } }
>    { exfalso; apply (Hneq p1). }
>  Qed.
>End eqdep_dec.





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