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Hi I am trying to do some mathematics in the Lean proof assistant to see how it works. I decided that it should be fun to play with idempotents of a commutative ring. Here's what I tried:

variables (A : Type) (R : comm_ring A)
definition KR : Type := \Sigma x : A, x * x = x

I then get the error 

failed to synthesize placeholder
A : Type,
x : A
⊢ has_mul A

So Lean seems to have forgotten that A is a ring?

So for example, if I change the definition to

definition KR (A : Type) (R : comm_ring A) :  Type := Σ x : A , x = x * x

then everything is fine. But this means that I have to carry extra bookkeeping data. Is there a way to use variables to get around the need to keep bookkeeping stuff around.

Jonathan Gallagher
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1 Answers1

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By default, Lean includes section variables and parameters only in definitions that actually use them. You can override this with the include and omit commands. But since comm_ring is a type class, you'll likely want to declare it as a class inference parameter anyway:

variables (A : Type) [comm_ring A]

Leaving out the parameter's name like this will include it in every definition by default, so with that your definition should work.

Sebastian Ullrich
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  • In Lean 3.3.1 and possibly earlier, the line after the one Sebastian Ullrich suggests should now probably be `definition KR : Prop := ...` (otherwise there are issues with a function from A to Type having type Type 1). – Kevin Buzzard Sep 30 '17 at 14:27