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From what I have read, eq_rect and equality seem deeply interlinked. Weirdly, I'm not able to find a definition on the manual for it.

Where does it come from, and what does it state?

Anton Trunov
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Siddharth Bhat
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1 Answers1

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If you use Locate eq_rect you will find that eq_rect is located in Coq.Init.Logic, but if you look in that file there is no eq_rect in it. So, what's going on?

When you define an inductive type, Coq in many cases automatically generates 3 induction principles for you, appending _rect, _rec, _ind to the name of the type.

To understand what eq_rect means you need its type, 

Check eq_rect.

here we go:

eq_rect
     : forall (A : Type) (x : A) (P : A -> Type),
       P x -> forall y : A, x = y -> P y

and you need to understand the notion of Leibniz's equality:

Leibniz characterized the notion of equality as follows: Given any x and y, x = y if and only if, given any predicate P, P(x) if and only if P(y).

In this law, "P(x) if and only if P(y)" can be weakened to "P(x) if P(y)"; the modified law is equivalent to the original, since a statement that applies to "any x and y" applies just as well to "any y and x".

Speaking less formally, the above quotation says that if x and y are equal, their "behavior" for every predicate is the same.

To see more clearly that Leibniz's equality directly corresponds to eq_rect we can rearrange the order of parameters of eq_rect into the following equivalent formulation:

eq_rect_reorder
     : forall (A : Type) (P : A -> Type) (x y : A), 
         x = y -> P x -> P y
Anton Trunov
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  • Thank you! What about the other two terms that are generated? I assume `_ind` is the induction principle. What is `_rec`? – Siddharth Bhat Jan 20 '18 at 13:12
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    Those principles have the same form, the difference between them lies in the type of predicate `P`: for `eq_rect` it's `A -> Type`, for `eq_ind` it's `A -> Prop`, and for `eq_rec` -- `A -> Set`. In fact, `eq_rec` is just a special case of `eq_rect`, which you can check with `Print eq_rec`. – Anton Trunov Jan 20 '18 at 13:24
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    Note that you can also generate such principles with things like `Scheme eq_rect := Minimality for eq Sort Type`, `Scheme eq_rec := Minimality for eq Sort Set`, `Scheme eq_ind := Minimality for eq Sort Prop`, and there are also dependent versions where you replace `Minimality` with `Induction` and get schemes like `forall (A : Type) (x : A) (P : forall a : A, x = a -> Type), P x eq_refl -> forall (y : A) (e : x = y), P y e`. I believe Coq automatically chooses `Minimality` for inductives in `Prop`, and `Induction` for inductives landing in `Set` and `Type`. – Jason Gross Jan 23 '18 at 16:04