## Stream: general

### Topic: f circ g = h or forall x, f (g x) = h x?

#### Kevin Buzzard (Apr 22 2018 at 23:05):

I am constantly proving that various functions are equal because they are both the unique function with some property. A lot of my proofs naturally prove f circ g = h. However Kenny often writes such goals as forall x, f (g x) = h x. Normally in Lean I would prove f circ g = h by applying funext and then intro x. However in the kind of mathematics I'm currently doing this is not the way to prove things -- in fact I hardly ever get my hands dirty with explicit terms, I'm just chasing around universal properties.

#### Kevin Buzzard (Apr 22 2018 at 23:06):

Kenny set up a structure with things like function.left_inverse g f and function.right_inverse g f and I had assumed these would unravel to g circ f = id etc but even these core functions evaluate to forall x, g (f x) = x

#### Mario Carneiro (Apr 22 2018 at 23:07):

The left inverse thing is convenient because that way you can write function.left_inverse g f x

#### Kevin Buzzard (Apr 22 2018 at 23:07):

What's a nice convenient way to deduce "forall x, g (f x) = h x" from "g circ f = h" ? I keep having to enter tactic mode and then rewriting the goal with rw, and usually rw \l.

#### Kevin Buzzard (Apr 22 2018 at 23:08):

Is there a proof of g circ f = h -> forall x, g (f x) = h x?

#### Kevin Buzzard (Apr 22 2018 at 23:08):

I am using it all the time

#### Mario Carneiro (Apr 22 2018 at 23:08):

then make it a lemma

fair do's

#### Kevin Buzzard (Apr 22 2018 at 23:09):

what's it called?

#### Mario Carneiro (Apr 22 2018 at 23:09):

it's a bit too specific for my taste, but if you fold it in with one of your definitions I'm sure it will work well

#### Kevin Buzzard (Apr 22 2018 at 23:09):

I think what I'd rather do is to rewrite all the definitions which we have which involve the x

#### Kevin Buzzard (Apr 22 2018 at 23:10):

however I am concerned that end users might need the "explicit" versions

#### Mario Carneiro (Apr 22 2018 at 23:10):

then have a special constructor for your more user-facing theorems that accepts the explicit version

#### Kevin Buzzard (Apr 22 2018 at 23:10):

I don't understand this

#### Mario Carneiro (Apr 22 2018 at 23:10):

I would make that lemma a biconditional BTW, you will want both directions

#### Kevin Buzzard (Apr 22 2018 at 23:10):

I know that 9 times out of 10 when i'm trying to prove f = g the first thing I do is apply funext

#### Kevin Buzzard (Apr 22 2018 at 23:11):

but here it's never true

#### Mario Carneiro (Apr 22 2018 at 23:11):

My initial thought on this is to use functions, you are basically doing category theory lite

right

#### Kevin Buzzard (Apr 22 2018 at 23:12):

I don't care about my actual definition of a localised ring

#### Kevin Buzzard (Apr 22 2018 at 23:12):

all I ever use is the universal property

#### Kevin Buzzard (Apr 22 2018 at 23:12):

which is a real relief

#### Kevin Buzzard (Apr 22 2018 at 23:12):

because the localised ring is a quotient type

#### Kevin Buzzard (Apr 22 2018 at 23:12):

and they are awkward to use

#### Mario Carneiro (Apr 22 2018 at 23:12):

So when do you actually need the x?

no idea

#### Kevin Buzzard (Apr 22 2018 at 23:13):

I am not sure I do

#### Mario Carneiro (Apr 22 2018 at 23:13):

At the very least you have some notion of a "user" that will use one of the theorems and prove a hypothesis by funext

#### Mario Carneiro (Apr 22 2018 at 23:13):

which theorems are most likely candidates for this?

#### Kevin Buzzard (Apr 22 2018 at 23:13):

I am not sure I have a clear notion of a user

#### Kevin Buzzard (Apr 22 2018 at 23:13):

I think I am the only user at the minute

hence the quotes

#### Kevin Buzzard (Apr 22 2018 at 23:14):

Maybe if Kenny does more exercises in Atiyah-Macdonald he'll become another user

#### Kevin Buzzard (Apr 22 2018 at 23:14):

Maybe I'll just rip out all the x's.

#### Mario Carneiro (Apr 22 2018 at 23:14):

or maybe your theorem has a conclusion that is a function equality, and the mythical user wants to congr_fun that equality

#### Mario Carneiro (Apr 22 2018 at 23:15):

you should identify the most outward-facing of your theorems and make them as pleasant to use as you can

#### Kevin Buzzard (Apr 22 2018 at 23:22):

This exercise (wanting a theory of localization because I have a use for it, but asking Kenny to prove the theorems) has really clarified for me how this whole process of writing a library works.

#### Patrick Massot (Apr 23 2018 at 07:57):

by applying funext and then intro x

Side remark: you can write funext x instead of funext, intro x. Soon you'll even be able to write ext x without thinking about whether you deal with an actual function or any other object for which extensionality makes sense

Last updated: May 10 2021 at 00:31 UTC