meeting -*- Outline -*-

* definitions (4.3)

  Q: Why are we worried about consistency when extending a theory?
  Q: What's a conservative extension?

** constant definition
------------------------------------------
        CONSTANT DEFINITION

Format:   
          ^
        c = t           (name)

 where t is a closed term

Effect:
  - adds new constant c to SG
  - adds new axiom |- c = t to rules

Example:
   ^                         (backward
 o = (\ f g x . f.(g.x))      function
                              composition)

Shorthand for functions:
     ^
id.x = x            (identity)
          ^ 
(f * g).x = g.(f.x) (forward composition)
          ^
(f o g).x = f.(g.x) (backward composition)

------------------------------------------
        Q: how is this like Haskell?
        Q: how are the types of constant definitions figured out?

** abbreviations (syntactic sugars)
*** local definitions
------------------------------------------
              LET

                  ^
(let v = t' in t) = (\ v . t).t'

Derived inference rules:

  ________________________   (vacuous let)
  |- (let v = t' in t) = t
  * if v is not free in t

  ___________________________ (let over \)
  |- (let v = t' in (\ y . t))
     = (\ y . let v = t' in t)
  * if y is not free in t'

------------------------------------------

        Q: Can you simplify  let x = 3 in x + x  ?