Using indexed types, we can restrict the form of legal values in some datatype.  A nice example is binary search trees, where we can statically enforce the binary search tree property using an indexed type bst l u, where l is a lower bound and u an upper bound on the data stored in the tree.

Basic idea of using singleton types like Nat n where n is a value from the index domain, to connect program expressions and index expressions.  The data value of type Nat n is a copy of n, but living in the syntactic category of program expressions.  This allows programs to operate on a proxy for n.  Singletons library in Haskell mentioned.

If indices to types come from a different syntactic category than programs, there are a few things you cannot do.  Some initial thoughts on how to work around these.

Indexed datatypes like vectors, where the indices come from a different syntactic category than program expressions.

For programs to make sense as proofs, they need to be terminating (cannot run forever), since otherwise you can write infinite loops that have any type.  Under Curry-Howard this means any formula is provable, which is one way to define inconsistency.  (And logics have to be consistent to be useful.)

If you have dependent types, classical reasoning, and the Curry-Howard isomorphism, you can write programs that look like they are invoking oracles for undecidable problems -- but they are not, and this is confusing.

CH can be applied to classical logic, too.  The seminal paper is A Formulae-as-Types Notion of Control by Timothy Griffin.  I discuss how backtracking implements the law of excluded middle.

Dependent types are discussed, particularly as used for expressing pre- and post-conditions of functions.

I follow up on some comments I made about Curry-Howard for first-order quantifiers in the previous episode.  Sheard's Omega language also mentioned (see links on his web page).  First-order quantifications turn into indexed types where the indices are not program expressions but come from another syntactic domain.

Discussion of the Curry-Howard isomorphism for the connectives of propositional logic (AND, OR, NOT, FALSE, IMPLIES).  Initial consideration of Curry-Howard for first-order and higher-order logic.  Dependent types.