=== 15-399 Recitation Section A ===

 Feb 2, 2005

1. Intuition behind neutral and normal proofs

 - Try to come up with your own intuition behind or interpretation of neutral 
   and normal proofs.

 - e.g.,

    |
  A v   : We have already proved A from hypotheses, and now want to exploit it
          in the most compact way.

    ^
  A |   : We don't have a proof of A yet, and now want to build its proof
          in the most compact way.

 - e.g., conjunction & introduction 

     ^     ^            
   A |   B |            This is the most compact way to build a proof of A & B.
  -----------[&-I]
          ^
    A & B | 

 For example, the following is not the most compact proof of A & B, 
 because we only have to prove A and B but we are unnecessarily proving 
 something more complicated (A & B) & C:


   (A & B) & C                We only have to prove A & B. Then why do 
  --------------[&-Eleft]     we bother with this more complicated proposition
            ^                 (A & B) & C? 
      A & B |

 - e.g., conjunction & elimination 

          |
    A & B v               This is one of the two most compact ways to exploit 
  ----------[&-E-left]    A & B. 
       |
     A v

3. Properties of neutral and normal proofs

 - A neutral proof is either
    using a hypothesis, or  
    applying an elimination rule to another neutral proof.

 - A normal proof is either
    a neutral proof, or
    applying an introduction rule to another normal proof.

Note that a neutral proof cannot exist without hypotheses. Hence you can't
build a neutral proof of A without introducing hypotheses in advance.
For example, you can't prove (because you begin with no hypothesis) :

          ^
  P \/ ~P | 

4. Normal proof of double negation of Peirce's law

5. Properties of constructive logic with & (conjunction), 
   => (disjunction), and T (truth).

  (1) Every normal proof is irreducible.  
      That is, a reducible proof is not normal (by contrapositive) 
      That is, if a proof contains a detour, it is not normal.

  The intuition is that if a proof contains a detour, it is not build in
  the most compact way and thus is not a normal proof. For example, consider:

    A      B
  ------------[&-I]
     A & B
    --------[&-E-left]
       A

  This is a detour because an introduction is followed immediately by an
  elimination rule. When you try to annotate it with arrows, you can't decide
  which arrow to use for A & B. If you look at the introduction rule part,
  you get:

      ^      ^
    A |    B |
  -------------[&-I]
           ^
     A & B |

  So, A & B is given an upward arrow. But, if you look at the elimination rule
  part, you get:

           |
     A & B v
    ---------[&-E-left]
         |
       A v

 But, you can't have: 

           ^
     A & B |
   -----------
           |
     A & B v

 So, you can't annotate any proof containig the above proof fragment.

  (2) Every non-normal proof is reducible.
      That is, every irreducbile proof is normal (by contrapositive).
      That is, if a proof does not contain a detour, it is normal.

6. From the two properties (1) and (2) above, we can always reduce a proof
  to a normal proof. 

  - First, check if your proof contains a detour. If it doesn't, it is normal
    and we are finished.
  - Otherwise, remove the detour and repeat the first step.

The normalization property states that for any proof, there exists a unique path
that leads to a normal proof. That is, you can find a unique sequence of 
detours that gives you a normal proof when removed.

The strong normalization property states that (not just the unique one but)
every sequence of detours lead to the same normal proof when removed.

7. Then, what happens when you add the disjunction connective \/?
  => See Assignment 4.