meeting -*- Outline -*-

* derivations (4.2)

  Q: what kind of practical problems might you how to if you try to use
  the derivation formats for reasonably sized realistic proofs?
      it's hard to type, you quickly run out of horizontal space,
      assumptions are repeated all over,
      explicitly invoking trivial rules like transitivity,
                 substitution, and symmetry


  ideas: write the proof sideways with indentation,
	 avoid repeating assumptions by using scope
	 
------------------------------------------
     BASIC CALCULATIONAL PROOF FORMAT

Translate
     D1  D2  ...  Dm
    _________________  {rule name}
      Phi |- t == t'

into:

  Phi
  |- t
  == { rule name }
      * [[D1]]
      * [[D2]]
      ...
      * [[Dm]]
  . t'

where [[Di]] is the translation of Di
------------------------------------------
        Scope of assumptions, Phi, is the whole thing
              - added assumptions shown in square brackets

------------------------------------------
         USING TRANSITIVITY

Translate
     DD1                     DDm
______________ {R1} ... ____________{Rm}
Phi |- t0 == t1          Phi |- tm-1 == tm
___________________________ {transitivity}
  Phi |- t0 == tm

into:

  Phi
  |- t0
  == { R1 }
      [[D1]]
  . t1
  == { R2 }
      [[D2]]
      ...
  . t2
  ...
  . tm-1
  == { Rm }
      [[Dm]]
  . tm

where [[Di]] is the translation of Di
------------------------------------------
        Q: what savings are achieved by this proof format?
           avoiding restatement of assumptions, and duplication of terms

        Q: How would you prove from arithmetic that
            (\ x . 2*x) == (\ y . y + y)?