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[Pierre Corbineau <Pierre.Corbineau AT imag.fr>]

This is how you do (Coq v8.2pl1).

Require Import JMeq.

Axiom UIP : forall (A:Type) (a:A) (e e':a = a) , e=e'.
Scheme real_JMeq_ind := Induction for JMeq Sort Prop.

Theorem JMeq_eq: forall (A:Type) (x y:A), JMeq x y -> x = y.
intros A x y H.
change (x = eq_rect A (fun t => t) y A (refl_equal A) ).
revert H.
refine ((_ : forall e,
JMeq x y ->  x= eq_rect A (fun t : Type => t) y A e ) (refl_equal A)).
revert y.
generalize A at 1 2 5 7.
intros B y e H.
revert e.
apply (real_JMeq_ind A x  (fun B y _ => forall e : B=A, x = eq_rect B 
(fun t => t) y A e)).
2:assumption.
intro e.
rewrite (UIP Type A e (refl_equal A)).
simpl.
apply refl_equal.
Qed.



Christian Doczkal a écrit :
> Hello
>
> I have a question about the JMeq Library
>
> It introduces the following axiom:
>
> Axiom JMeq_eq : forall (A:Type) (x y:A), JMeq x y -> x = y.
>
>
> In [1] McBride says that the elimination principles on can derive using
> this axiom "give[s] exactly the same strength as Martin-Löfs equality
> extended with Altenkirch and Streicher's 'uniqueness of identity proofs'
> axiom."
>
> So is JMeq_eq is provable in Coq using UIP? Does anyone have a proof?

--
Pierre Corbineau          | 
Pierre.Corbineau AT imag.fr
VERIMAG - Centre Équation | Tel: (+33 / 0) 4 56 52 04 42
2, avenue de Vignate      | Office nr B7
38610 GIÈRES - FRANCE     | http://www-verimag.imag.fr/~corbinea/
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