How to convert a recipe to a cold proof

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Following a recipe that calls for 1st proof of 40 minutes followed by shaping and another 40 minute proof. Assuming I'd like to use a cold ferment, how would I "convert" this?

Those are very short proofing times.. is this recipe yeasted or sourdough? Either way, how much “leavener” is in this recipe?

One suggestion is to use 1/8 tsp yeast in a poolish then another 1/4 tsp in the main dough, per 1kg flour weight.  This is the amount of yeast I use in my standard recipe.  I use the same amount of yeast cold proofed and room temperature proofed.  To me it doesn't matter, there is alot of lattitude in bulk fermenting time.  Final proofing time is visual and also has wide lattitude. 

Poolish is a different method to bake bread.  You are using a "straight dough" method.   You appear to be mixing all ingredients, bulk fermenting (first proof), shaping, then proofing (second proof).  Using a poolish means making the starter (the poolish), letting it mature overnight, mixing the dough, bulk fermenting (first proof) then shaping and proofing (second proof). 

You can replace the bulk ferment (first proof) in your method with a cold proof.  I would reduce the yeast in your recipe by alot, maybe 50% or more, but that is my subjective preference. 

Are you using the search function on this site?  Or using Google to search?  These search methods are very helpful in baking. 

Thanks. To be honest you folks have been so helpful I got a little lazy. Also asking here allows for follow-up questions and more recipe-specific instructions. But you're right, I should search more often.