Coq has a type Prop of proof irrelevant propositions which are discarded during extraction. What are the reason for having this if we use Coq only for proofs. Prop is impredicative, so Prop : Prop, however, Coq automatically infers universe indexes and we can use Type(i) instead everywhere. It seems Prop complicates everything a lot.
I read that there're philosophical reasons for separating Set and Prop in Luo's book, however, I didn't find them in the book. What are they?
$\mathtt{Prop}$ is very useful for program extraction because it allows us to delete parts of code that are useless. For example, to extract a sorting algorithm we would prove the statement "for every list $\ell$ there is a list $k$ such that $k$ is ordered and $k$ is a permutatiom of $\ell$". If we write this down in Coq and extract without using $\mathtt{Prop}$, we will get:
sort which takes lists to lists,verify which runs through $k$ and checks that it is sorted, andpi which takes $\ell$ to $k$. Note that pi is not just a mapping, but also the inverse mapping together with programs verifying that the two maps really are inverses.While the extra stuff is not totally useless, in many applications we want to get rid of it and keep just sort. This can be accomplished if we use $\mathtt{Prop}$ to state "$k$ is ordered" and "$k$ is a permutation of $\ell$", but not "for all $\ell$ there is $k$".
In general, a common way to extract code is to consider a statement of the form $\forall x : A \,.\, \exists y : B \,.\, \phi(x, y)$ where $x$ is input, $y$ is output, and $\phi(x,y)$ explains what it means for $y$ to be a correct output. (In the above example $A$ and $B$ are the types of lists and $\phi(\ell, k)$ is "$k$ is ordered and $k$ is a permutation of $\ell$.") If $\phi$ is in $\mathtt{Prop}$ then extraction gives a map $f : A \to B$ such that $\phi(x, f(x))$ holds for all $x \in A$. If $\phi$ is in $\mathtt{Set}$ then we also get a function $g$ such that $g(x)$ is the proof that $\phi(x, f(x))$ holds, for all $x \in A$. Often the proof is computationally useless and we prefer to get rid of it, especially when it is nested deeply inside some other statement. $\mathtt{Prop}$ gives us the possibility to do so.
Added 2015-07-29: There is a question whether we could avoid $\mathsf{Prop}$ altogether by automatically optimizing away "useless extracted code". To some extent we can do that, for instance all code extracted from the negative fragment of logic (stuff built from the empty type, unit type, products) is useless as it just shuffles around the unit. But there are genuine design decisions one has to make when using $\mathsf{Prop}$. Here is a simpe example, where $\Sigma$ means that we are in $\mathsf{Type}$ and $\exists$ means we are in $\mathsf{Prop}$. If we extract from $$\Pi_{n : \mathbb{N}} \Sigma_{b : \{0,1\}} \Sigma_{k : \mathbb{N}} \; n = 2 \cdot k + b$$ we will get a program which decomposes $n$ into its lowest bit $b$ and the remaining bits $k$, i.e., it computes everything. If we extract from $$\Pi_{n : \mathbb{N}} \Sigma_{b : \{0,1\}} \exists_{k : \mathbb{N}} \; n = 2 \cdot k + b$$ then the program will only compute the lowest bit $b$. The machine cannot tell which is the correct one, the user has to tell it what they want.
$\mathrm{Prop}$ is impredicative, which create a very expressive proof system. However it is "too" expressive in the following sense:
$$ \mathrm{impredicative\ Prop} + \mathrm{large\ elimination} + \mathrm{excluded\ middle} $$
is inconsistent. Usually you want to keep the possibility to add the excluded middle, so one solution is to keep large elimination and make Prop predicative. The other is to suppress large elimination.
Coq did both! They renamed the predicative Prop to Set, and disabled large elimination in Prop.
The expressiveness gained by impredicativity is "reassuring" in the sense 99% of "reasonable" mathematics can be formalized with it, and it is known to be consistent relative to set theory. This makes it likely they won't weaken it to something like Agda, which only has predicative universes.
Prop : Prop, that would be inconsistent. Rather quantifications over all propositions is again a proposition.
$\endgroup$
Even if you are not interested in extracting programs, the fact that Prop is impredicative allows you to build some models which you can't build using a predicative tower of universes. IIRC Thorsten Altenkirch has a model of System F using Coq's impredicativity.
The principle of propositions-as-types (or formulas-as-types), also known as the Curry-Howard correspondence, is the key idea for viewing (intuitionistic) type theories as logical systems and to apply type theories (and constructive mathematics in general) to computer science. Suppose U is the type of all propositions, then U is also the type of all types by the identification. Therefore, type U of all types naturally occurs and, in particular, it is the type of itself. In Martin-Löf's (predicative) type theory, propositions and types are still identified, but the type of all types is replaced by an infinite sequence of type universes U0 U 1: U2..... Hence, one can not quantify over all propositions or predicates, although quantification over each universe is allowed.
That's the reason type and prop are separated. Without this separation the resulting system becomes inconsistent.
