(* Ex 3.11.7, allEven (in class) *)
fun allEven([]) = true
 (*  | allEven(a::rest) = if a mod 2 = 0 then allEven(rest) else false; *)
  | allEven(a::rest) = (a mod 2 = 0) andalso allEven(rest); 




(* Ex 3.11.8, hasEven (in class) *)
fun hasEven([]) =  false
  (* | hasEven(a::rest) = if a mod 2 = 0 then true else hasEven(rest); *)
   | hasEven(a::rest) = (a mod 2 = 0) orelse hasEven(rest);

(* Ex 3.11.9, notAllEven (in class) *)
fun notAllEven([]) = false 
  | notAllEven(a::rest) = if a mod 2 = 1 then true else notAllEven(rest);


(* Ex 3.11.10, noneEven (in class) *)
fun noneEven([]) = true
  | noneEven(a::rest) = if a mod 2 = 0 then false else noneEven(rest);


(* From Section 3.13 *)
(* Wrong version! *)
fun allHaveDivisor([], yy) = true 
  | allHaveDivisor(xx, []) = false
  | allHaveDivisor(x::xRest, y::yRest) =  
       if x mod y = 0      
       then allHaveDivisor(xRest, y::yRest)
       else allHaveDivisor(x::xRest, yRest);

(* Correct version *)
fun allHaveDivisor([], yy) = true
  | allHaveDivisor(x::xRest, yy) =
       let fun hasDivisor([]) = false
             | hasDivisor(y::yRest) = if x mod y = 0 
                                     then true
                                     else hasDivisor(yRest);
       in
           hasDivisor(yy) andalso allHaveDivisor(xRest, yy)
       end;

(* Ex 3.13.1, allLessThan (for all, for all) (in class) *)
fun allLessThan([], yy) = true
  | allLessThan(x::xRest, yy) = 
      (* lessThanAll asks the question "is x less than all things in this list?" *)
    let fun lessThanAll([]) = true
          | lessThanAll(y::yRest) = if x < y then lessThanAll(yRest) else false; 
    in
        lessThanAll(x) andalso allLessThan(xRest, yy)
    end;


(* Ex 3.13.3, allHaveAddInv (for all, there exists) (in class) *)
fun allHaveAddInv(original) =
    let fun hasAddInv(x, []) = false
          | hasAddInv(x, y::yRest) = x + y = 0 orelse hasAddInv(x, yRest);
      fun allHaveAddInvHelper([]) = true
        | allHaveAddInvHelper(x::xRest) = 
             hasAddInv(x, original) andalso allHaveAddInvHelper(xRest)
    in
        allHaveAddInvHelper(original)
    end;


(* Ex 3.13.4 (there exists, for all) (HW) *)
fun hasDivisorOfAll(original) =
    let
        fun isDivisorOfAll(x, []) = ??
          | isDivisorOfAll(x, y::yRest) = ??
        fun hasDivisorOfAllHelper([]) = ??
          | hasDivisorOfAllHelper(x::xRest) =
            isDivisorOfAll(x,original) orelse hasDivisorOfAllHelper(xRest)
    in
        ??
    end;

(* Ex 3.13.5 (there exists, there exists) (HW) *)
fun hasCommonElement([], bb) = ??
   | hasCommonElement(a::aRest, bb) =
       let 
          fun containsA([]) = ??
             | containsA(b::bRest) = ??
        in
           ??
        end;