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Pairs - <a,b>
Nuprl's standard pair forming operation is <a,b>. The basic destructor for pairs, sometimes called
spread, supplies a pair's components to a function, and is expressed as an operator binding two variables
which evaluates thus:
<a,b>/x,y. f(x;y) * f(a;b)
The projection functions are special cases:
p.1 is p/x,y. x p.2 is p/x,y. y
<a,b>.1 * a <a,b>.2 * b
The "cartesian product" type AB comprises pairs <a,b> where aA and bB. This is really just a special case of u:AB(u) in which variable u does not occur free in the second type expression. The type u:AB(u) comprisese pairs <a,b> where aA and bB(a). The expression u:AB(u) denotes a type when A denotes a type and, further, B(u) denotes a type-valued function in u:A. Thus, if A denotes the empty type, then u:AB(u) does too, no matter what B(u) is (so long as no variables other than u are free in B(u)).
This type constructor is often called the "general sum" of a family of types, and indeed the most common notation for it is "u:A. B(u)".
Here's a theorem about pairing projections, complementing the above computational facts about projecting off of pairs:
We adopt the convention for forming standard tuples by iterating through the second component, so the triple <a,b,c> is a pair having <b,c> as its second component. The notation for spreading tuples is extended as well.