Question

I am currently working through the book Software Foundations. When theorems are defined, I often see chains of implications where I believe conjunctions would make more sense. For example, in defining the pigeonhole principle, the authors write:

Theorem pigeonhole_principle: ∀{X:Type} (l1 l2:list X),
  excluded_middle →
  (∀x, appears_in x l1 → appears_in x l2) →
  length l2 < length l1 →
  repeats l1.

This definition would make more sense to me if it looked more like this:

Theorem pigeonhole_principle: ∀{X:Type} (l1 l2:list X),
  (excluded_middle /\
  (∀x, appears_in x l1 → appears_in x l2) /\
  length l2 < length l1) →
  repeats l1.

Why is the first version correct? Why wouldn’t conjunctions be more appropriate?

That is just an example. More generally, I am asking why conjunctions seem to be shunned in favor of chains of implications in coq.

Answer 1

This is the theorem-prover version of function currying.

To summarise, the two expressions are both correct (they are equivalent).

One says: Give me an “excluded middle”. Now give me hypothesis 2. Now give me hypothesis 3. Okay, here is a property “repeats l1”.

The other says: Give me an “excluded middle”, and hypothesis 2, and hypothesis 3. Okay, here is a property “repeats l1”.

Users and implementers of proof assistants such as Coq are also functional programmers and the curried version is as natural for them as the uncurried one.