I'll use the new syntax for local variable declaration

Recall the example from class


Consider c = var x=True in
              ( (var x=1 in
                  (while x < 1 do x:= x+1)) ;
                x:= False)

Assume G= {}

We derive G |- c : Cmd       


Here's the proof tree:   
                                                              G'' |- x : Int
							      G'' |- 1 : Int
                                                        (Op) -----------------
                                    G'' |- 1 : Int           G'' |- x+1 : Int
                                    G'' |- x : Int           G'' |- x : Int
                             (Cmp) ---------------  (Assign) -----------------
                                  G'' |- x<1 : Bool   G'' |- x:= x+1 : Cmd
                         (While) -----------------------------------
                                 G'' |- while x<1 do x:= x+1
                 G' |- 1 : Int                  
                                  G''=G', x:Int={x : Int}
         (Newvar) ---------------------------------------   
                       G' |- (var x=1 in                    (Var) G' |- x : Bool 
                              (while x < 1 do x:= x+1))
                       
             (Seq)   ----------------------------------------------------
                               G' |- (var x=1 in
                                      (while x < 1 do x:= x+1)) ;
                                      x:= False)
                               
            G |- True : Bool   G'= G,x:Bool = {x:Bool}
(Newvar)  -----------------------------------------------
             G |- var x=True in
                  ( (var x=1 in
                     (while x < 1 do x:= x+1)) ;
                    x:= False)



What about?

(2)  G = { x : Int, y : Bool}
     c = var x:= true; (y:= x /\ true)

(3)  G = { x : Int, y : Bool}
     c = x:= true; (y:= x /\ true)

(4)  G = {}
     c = var x:=1; (var x:= x <1; x:= x /\ True)