CONSTRUCTIVE LOGIC

(1) MINIMAL IMPLICATIONAL LOGIC (N^>_m)

Propositions: A ::= a | A => A
Contexts: Gamma = finite set {xi:Ai} with distinct xi
Sequent: Gamma |- A  (premise,conclusion)
Rules:

Examples: ((A > (B > C)) > ((A > B) > (A > C)))
          A > ((A > (B > C)) > ((A > B) > (A > C)))

	- use contexts
	- use tagged assumptions

Redundancy:

	    D1
	G,x:A |- B
        ----------        D2
        G |- A > B	G |- A
	----------------------
		G |- B

Construct proof of G |- B without using x:A
Leaves of the form 
	G,Del,x:A,y:C|-C or G,Del,x:A|-A

Replace leaves G,Del,x:A|-A with D2 by adding
Del to the premise of each sequent in Delta
(May need to rename variables in D2)
Delete x:A from premise of every sequent
Resulting Deduction: D1[D2/x]

NOTE: not all assumptions are necessarily used
      some assumptions used more than once

This transformation is called Reduction or Normalization
step.

Curry/Howard Isomorphism (Propositions-as-Types)

TypeChecking \lambda^>:

Sequent: G |- M:A  M=term representing a deduction of A
Rules

Redundancy Example:  (\x:A.M)N:B ==> M[N/x]

D1 = term M   D2 = term N
Reduction step = Beta


ADDING CONJUNCTION, DISJUNCTION, NEGATION
(INTUITIONISTIC PROPOSITIONAL LOGIC)

Propositions:
  A ::= a| A=>A | A ^ A | A v A | bot
  (negation is encoded: ~A == A => bot)

^-elim and intro
v-elim and intro
bot-elim: G |- bot
          --------
          G |- A

A^B => B^A,  AvB => BvA, A^(B^C) => (A^B)^C

Minimal Propositional Logic - drop bot-elim
Classical Propositional Logic: Add negation
to minimal logic:

	G,x:~A |- bot
	-------------  (by-contra)
	   G |- A

In Classical logic, bot-elim can be DERIVED:
  if G |- bot, then G,x:~A |- bot, hence G|-A


Note: A => ~~A