-----------------
Deductive Systems
-----------------

-- a deductive system has 2 parts:
   1. definition of judgment form or collection of judgment forms
      * judgment = assertion, fact that may or may not be true
      * a valid or true judgment
   2. collection of rules
      * axioms
      * inference rules

      J1 ... Jn
      --------- name
          J

-- rules include some number of premises above the line and a conclusion
below the line.  We often put a name beside the rule to allow us to refer 
to it easily.

-- rules often contain "metavariables" -- stand for classes of objects
and may be instantiated by any object in the class.

-- a judgment is valid if:
   1. it is the conclusion of an axiom
   2. it is the conclusion of a rule where all premises are valid judgements

-- a derivation is a proof a judgement is valid.  Hence a derivation is
   1. either an axiom, or
   2. a rule plus derivations for all of the premises of the rule.

-- examples:

1. |- N nat     "N is a natural number"

                           |- N : nat
----------- Z            --------------- S
|- Z : nat                 |- S N : nat


Note:  N is a "meta-variable".  There are many instances of rule S.

A derivation proving that S (S (S Z)) is a natural number:

--------- Z
|- Z nat
------------------  S
|- S Z nat
------------------  S
|- S (S Z) nat
------------------  S
|- S (S (S Z)) nat


2. |- add N1 N2 = N3       "N1 plus N2 equals N3"


                                 |- add N1 N2 = N3
------------------- addZ       --------------------------- addS
|- add Z N = N                   |- add (S N1) N2 = S N3


3. |- even N              "N is an even number"
   |- odd N               "N is an odd number"


---------- evenZ
|- even Z  


|- odd N                   |- even N
------------- evenS       ------------- oddS
|- even (S N)              |- odd (S N)           


Alternative definition:

---------- even2Z         
|- even2 Z           


|- even2 N       
-----------------  even2S 
|- even2 (S (S N))



---------- odd2Z         
|- odd2 (S Z)           


|- odd2 N       
-----------------  odd2S 
|- odd2 (S (S N))


-----------------
Rule Induction
-----------------

-- the given inference rules are an *exhaustive* source of information
about the valid judgments.
-- no judgment is valid unless you can derive a proof for it using 
some finite number of the given inference rules

-- this leads to a proof principal (Rule Induction):

   * to show every derivable judgment has some property P, show for every
     rule in the deductive system:

      J1 ... Jn
      --------- name
          J

     if J1 ... Jn have property P then J has property P

-- examples:

1. Given a property P, we know that P is true for all natural numbers, 
if we can prove:

   * P holds unconditionally for Z.  Corresponds to rule:

                ----------- Z        
                 |- Z : nat                

   * Assuming P holds for N then P holds for (S N).  Corresponds to rule:

                   |- N : nat
                ------------- S
                  |- S N : nat

-- also called "induction on the structure of natural numbers" or more generally
"structural induction" when the objects are numbers, but other things like trees
or programming language expressions.

Theorem:  For all natural numbers N, one of the following judgements
is derivable |- even N  or |- odd N.

Proof:

By induction on the derivation of |- N nat.  
(Or "by induction on the structure of naturals N")

Case:  

--------
|- Z nat 


|- even Z     (by rule evenZ)


case:

 |- N nat
-------------
 |- S N nat


(1) |- even N    or   (2) |- odd N        (by IH)

Assuming (1):
|- odd S N                                (by (1) and oddS)

Assuming (2):
|- even S N                               (by (2) and evenS)

QED


Theorem:  For all N1, N2, there exists N3 such that |- add N1 N2 = N3.
 ie: add is a total relation.

Proof:
By induction on the structure of N1.

case:

--------
|- Z nat 


|- add Z N2 = N2     (by addZ)

case:

 |- N nat
------------- succ
 |- S N nat


|- add N N2 = N3'        (by IH)
|- add (S N) N2 = S N3'  (by addS)

QED


Theorem:  If |- even2 N then |- even N.
Proof:
By induction on the derivation |- even2 N.

case:

---------- even2Z         
|- even2 Z           

|- even Z             (by evenZ)

case:

|- even2 N       
-----------------  even2S 
|- even2 (S (S N))


|- even N            (by IH)
|- odd (S N)         (by oddS)
|- even (S (S N))     (by evenS)

QED