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[t x <txrev319 AT gmail.com>]

Cedric: You're right, the lemma I stated is false.

Araud: Thanks, that's exactly what I was going for.

On Fri, Sep 6, 2013 at 7:09 AM, Cedric Auger <sedrikov AT gmail.com> wrote:

2013/9/6 Arnaud Spiwack <aspiwack AT lix.polytechnique.fr>

forall X:Type, forall (A B:X->Prop), (forall x, A x <-> B x) -> forall P:X->Prop, exists x, A x /\ Px <-> exists x, B x /\ P x.

On 5 September 2013 22:20, t x <txrev319 AT gmail.com> wrote:

[We're using propositional equivalence through all this.]

Here is a simple lemma. (I don't need help proving it. I need help making it more abstract).

(1)

  Lemma lem__Z_split:
    forall (l r: Z) (P: Z -> Prop),
    (exists i, l <= i < r /\ P i) =
    (P l \/ (exists i, l +1 <= i < r)).

Using propositional equivalence, this can be proved easily using the Z destructuring of {a = b} + {a < b} + {a > b}.

easily ? That clearly seems wrong to me (assuming that "=" should be replaced with "<->")

Take :

l=0
r=2
P=λ x. ⊥ (P is the empty set)

Then "∃ i. l≤i<r ∧ P i" is obviously wrong, but
"P l ∨ ∃ i. l+1≤i<r" is obviously right (by dropping the left part and choosing i=1).

 

However, I want to formulate a more general theorem, something like this:

(2)

  forall (A B : Sets) (*  -- not necessairly the same "Set" as Coq's "Set" *),

    A = B (* in the sense that for all x, x in A <-> x in B *),

      exists x, x in A /\ P x = exists x, x in B /\ P x

Then, all I need is to show that the "set" defined by l <= i < r is equiv to teh set defined by i = l \/ (l+1 < i < r).

I guess you mean "i = l ∨ (l+1 ≤ i < r)"

And if you want sets, I guess you could start with "Definition set X := X -> Prop.".

Definition belongs {A:Type} (x : A) (S : set A) := S x.

Notation "x ∈ P" := (belongs x P) (at level …).

Then you would have:

Definition equiv_sets {A : Type} (S1 S2 : set A) :=
  forall x, x ∈ S1 <-> x ∈ S2.

Notation "S1 ≡ S2" := (equiv_sets S1 S2) (at level …).
 

I'm having trouble formulating (2) in Coq. Can someone show me sample code (or point me at some chapter I should read?)

Thanks!

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.../Sedrikov\...