% generates proof directory "exercise1a"
newcontext "exercise1a";

% a type
T: TYPE;

% predicates
p: T -> BOOLEAN;
q: (T,T) -> BOOLEAN;
r: T -> BOOLEAN;
s: T -> BOOLEAN;

% a function
f: T->T;

% formulas to be proved

A: FORMULA
     (EXISTS(x: T): p(x)) AND
     (FORALL(x: T): p(x) => (EXISTS(y: T): q(x, y))) =>
       (EXISTS(x, y: T): q(x, y));

B: FORMULA
     (FORALL(x: T): p(x) OR r(x)) AND
     (FORALL(x: T): p(x) => q(x, x)) AND
     (FORALL(x: T): r(x) => q(x, f(x))) =>
       (FORALL(x: T): (EXISTS(y: T): q(x, y)));
 
C: FORMULA
    (FORALL(x: T): p(x) => r(x)) AND
    (FORALL(x: T): r(x) => (EXISTS(y:T): q(x, f(y)))) AND
    NOT(EXISTS(x: T, y: T): q(x, y)) => 
      (FORALL(x: T): NOT p(x));