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Re: [Coq-Club] proof uninformativeness vs. proof irrelevance


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  • From: Jonathan Leivent <jonikelee AT gmail.com>
  • To: coq-club AT inria.fr
  • Subject: Re: [Coq-Club] proof uninformativeness vs. proof irrelevance
  • Date: Sun, 24 Apr 2016 10:12:39 -0400
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On 04/24/2016 12:22 AM, Abhishek Anand wrote:
Your proof essentially shows that Erasable_inj implies proof relevance of
Foo, and is thus inconsistent with assuming proof irrelevance of Foo :
(No was a Yes? Sorry if I am misunderstanding something.)

I believe that is correct.



Inductive Erasable(A : Set) : Prop :=
erasable: A -> Erasable A.

Arguments erasable [A] _.

Axiom Erasable_inj : forall {A : Set}{a b : A}, erasable a=erasable b ->
a=b.

Inductive Foo : Prop :=
foo: nat -> Foo.

Definition foo2erasable(f : Foo) : Erasable nat :=
match f with
| foo x => erasable x
end.

Lemma FooRelevant : foo 1 <> foo 2.
Proof.
intros H. apply f_equal with (f := foo2erasable) in H.
simpl in H.
apply Erasable_inj in H.
discriminate H.
Qed.


Do you need to assume Erasable_inj in its full generality? or just some
specific instances for some sets?
If only certain instances suffice, and if there was a way to easily
understand the implications of such instances, it may still be possible to
safely have some instances of proof irrelevance.
More formally, if we assume @Erasable_inj A for some set A, is there are a
simple characterization of the Props (members of sort Prop) whose proof
relevance become implied?


Unfortunately, one of the more important cases of Erasable_inj I use is when the Set is list A for arbitrary A. That arbitrary A would seem to imply that there is no way to restrict this proof-relevance domino effect.

Thanks for the help.

-- Jonathan




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