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[Cyprien Mangin <cyprien.mangin AT m4x.org>]

You need a form of predicate extensionality:

forall (T U : Type) (P Q : T -> U -> Prop), (forall x y, P x y <-> Q x y) -> P = Q

As remarked in https://coq.inria.fr/cocorico/CoqAndAxioms, this is entailed by propositional extensionality + functional extensionality.

On Wed, May 11, 2016 at 7:36 PM, scott constable <sdconsta AT syr.edu> wrote:

Hi Cyprien,

I realize that I can prove an IFF, but what I really need for my theorems is a Leibniz equality. I tried functional extensionality without any success. I'm not sure what other types of extensionality would be applicable.

~Scott

On Wed, May 11, 2016 at 1:32 PM, Cyprien Mangin <cyprien.mangin AT m4x.org> wrote:

Hello,

You will need some kind of extensionality to prove exactly your lemma. What you can prove is the following:

∀ (R : U → V → Prop) (Q : V → T → Prop) (P : T → W → Prop) x y, (P ∘ Q ∘ R) x y  ↔ ((P ∘ Q) ∘ R) x y.

--

Cyprien

On Wed, May 11, 2016 at 7:07 PM, scott constable <sdconsta AT syr.edu> wrote:

Correction: "I think it should be possible to prove associativity of my definition of composition"...

On Wed, May 11, 2016 at 1:06 PM, scott constable <sdconsta AT syr.edu> wrote:

Thanks for the quick response Frédéric, but I already have a large project which relies heavily on my definition of composition, and the CoLoR library uses a different definition of composition. Hence I would have to rework a lot of my project to fit the CoLoR definition. I think it should be possible to prove associativity of my definition of relation in Coq, and I would like to know how to do so.

~Scott

On Wed, May 11, 2016 at 1:00 PM, Frédéric Blanqui <frederic.blanqui AT inria.fr> wrote:

Hello. You will find useful developments on relations in CoLoR.Util.RelUtil.v (coq-color on opam). See http://color.inria.fr/doc/CoLoR.Util.Relation.RelUtil.html for the definitions and theorems without the proofs. Best regards, Frédéric.

Le 11/05/2016 18:51, scott constable a écrit :

Hi All,

I have the following definition of composition:

Inductive Compose : T → V → Prop :=
| comp_intro : ∀ x y z, R1 x y → R2 y z → Compose x z.

Notation "P ∘ R" := (Compose R P) (at level 55, right associativity).

And I am hopelessly lost as to how to prove this theorem:

Lemma compose_assoc : ∀ (R : U → V → Prop) (Q : V → T → Prop)
                          (P : T → W → Prop), P ∘ Q ∘ R = (P ∘ Q) ∘ R.

Any help would be extremely appreciated!

Thanks in advance,

~Scott Constable