(t is little tau, T is large tau, E is 'in', and + is that other symbol)
-[Eq, Neq]
+[Eq, NEq]
-e:bool <= e1:t & e2:t
-where e is e1 Eq e2.
+T+ e1 Eq e2:bool <= T+ e1:t & T+ e2:t
+
+If e1 and e2 are of the same type, then Eq or NEq return bool
[Plus, Minus, Times, Div]
-e:int <= e1:int & e2:int
-where e is e1 Plus e2.
+T+ e1 Plus e2:int <= T+ e1:int & T+ e2:int
+
+The operators Plus/Minus/Times/Div return int if both operands are ints
[Lt, ELt, Gt, EGt]
-e:bool <= e1:int & e2:int
-where e is e1 Lt e2.
+T+ e1 Lt e2:bool <= T+ e1:int & T+ e2:int
+
+The operators Lt/ELt/Gt/EGt return bool if both operands are ints
[Assignment]
T+ i := e:t <= i:t E T & T+ e:t
-where the assignment is identifier i = expression e.
-
-[ExpT]
-
-u,e:t <= e:t & u:t
-where the expression is type u expression e.
+The assignemnt of e to i returns type t if both i and e have type t.
[ENeg]
-e:int <= e:int
+T+ ENeg e:int <= T+ e:int
+ENeg e returns int if e is of type int
[ENot]
-e:bool <= e:bool
+T+ ENot e:bool <= e:bool
+
+ENot e returns bool if e is of type bool
+
+[EVar]
+
+T+ i:t <= i:t E T
+
+i has type t if i is defined in the context with type t.
+
+[EInt]
+
+T+ n:int
+
+n has type int
+
+[EBool]
+
+T+ b:bool
+
+b has type bool
+
+[EReadI]
+T+ n:int
-[SExp, SBlock]
+EReadI returns an int
-S:NoType <= e:t
+[EReadB]
+
+T+ b:bool
+
+EReadB returns a bool
+
+[EPost]
+
+T+ EPost i:int <= i:int E T
+
+EPost i is of type int if i is defined in T with type int.
+
+[SExp]
+
+T+ e <= T+ e:t
+
+[SBlock]
+
+T+ s;SBlock ss <= T+ s => T' , T'+ ss => T''
+
+the first statment s, in the block, is typechecked in the context T and returns the context T', the rest of the block is then recursively typeckecked in the context T'
[SIf]
T+ if e then s1 else s2 <= T+ e:bool & T+ s1 & T+ s2
+if e is of type bool and s1 and and s2 typechecks in the context T, then the same context is returned
+
[SWhile]
T+ while e do s <= T+ e:bool & T+ s
+If e is of type bool and s typechecks in context T then the same context is returned
+
+
[SDecl]
-T+ i:t => T', i:t <= i!ET & e:t & u:t
+T+ t i = e => T,i:t <= i!ET & e:t
+
+if i and e are of the same type and i is not declared in the current scope then i is added with type t to the context.
+
+[SDeclD]
-(Type u Ident i = Exp e)
+T+ t i => T,i:t <= i!ET
+if i is not declared in the current scope, then i is added to the context with type t
+[SNoop]
+T+ s
+SNoops does nothing so the same context is returned
+[SPrint]
+T+ e <= T+ e:t
+if e has type t then SPrint returns the same context