meeting_050905

Pattern matching revisited
--------------------------

  Don't allow

    f A 0 (nil A)

  But still allow

    f (s m) (cons m x xs)

  Always bind to the first m.

  Decision: don't implement this at first. No pattern matching at all on
  inductive families.

Meta variables
--------------

  Do not identify communication points and meta variables.

El and universes
----------------

  Abstract syntax for values

    data Value = ...
    data Type  = El Value Sort
	       | Pi Type Type
    data Sort  = Prop | Set Nat

  Rules for the subtyping with coercions:

    Γ, x:V ⊢ x : V ≤ El A --> v'   Γ, x:V ⊢ app v v' : El (B v') ≤ W --> w
    ----------------------------------------------------------------------
                   Γ ⊢ v : El (πAB) ≤ ∏x:A.B --> λx. w

    Γ, x:El A ⊢ x : El A ≤ V --> v'   Γ, x:El A ⊢ v v' : W ≤ El (B v') --> w
    ------------------------------------------------------------------------
                   Γ ⊢ v : ∏x:A.B ≤ El (πAB) --> lam (λx. w)

  The problem: We don't have eta.
    app (lam f) == f, but
    lam (app f) != f

  Thierry says this might not be easy to get.

  Decision: Stick to what we know we can get.

    Type level definitions only with datatypes:

    data Ref (R : El A -> El A -> Set) : Set where
      mkRef : ((x : El A) -> R x x) -> Ref R

    data Rel (A : Set) : Set_1 where
      mkRel : (El A -> El A -> Set) -> Rel A

    Wrapping and unwrapping have to be done explicitly.  This is slightly
    better than the alternative definition of Rel (with the small pi) since
    we only have to unwrap once instead of unwrapping each application.

  Core type judgement

      M : Set_i
    -------------
    El_i M type_i

    A type_i    B type_i [x:A]
    --------------------------
	(x:A) -> B type_i

    A type_j   j < i
    ----------------
       A type_i

    --------------
    Set_i type_i+1

Mutual inductive recursive definitions
--------------------------------------

  Definitions:

    x1 : A1 = M1
    .
    .
    .
    xn : An = Mn

  Type check

    (x1:A1)...(xn:An), and then
    M1:A1
    M2:A2 (x1=M1)
    ...
    Mn:An (x1=M1, x2=M2, ..)

  For datatypes, constructors are only visible in the rhs of the other
  definitions (not in the types). In other words we treat the constructors
  as the _definition_ of the datatype.

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