var S, T : Type

types Bool;

Pred : -Type -> Type := \ S : -Type . S ->? Unit;

Set : Type -> Type := \ S : Type . S ->? Unit

ops True, False : Bool

ops emptySet : Set S;

{__} : S -> Set S;

__isIn__ : S * Set S ->? Unit;

__subset__ : Pred (Set (S) * Set (S));

__union__, __intersection__, __\\__ : Set S * Set S -> Set S;

__disjoint__ : Pred (Set (S) * Set (S));

__*__ : Set S * Set T -> Set (S * T);

__disjointUnion__ : Set S * Set S -> Set (S * Bool);

inl, inr : S -> S * Bool

forall x, x' : S; y : T; s, s' : Set S; t : Set T

. not x isIn emptySet

. x isIn { x' } <=> x = x'

. x isIn s <=> s x

. s subset s' <=> forall x : S . x isIn s => x isIn s'

. x isIn (s union s') <=> x isIn s \/ x isIn s'

. x isIn (s intersection s') <=> x isIn s /\ x isIn s'

. x isIn (s \\ s') <=> x isIn s /\ not x isIn s'

. s disjoint s' <=> s intersection s' = emptySet

. (x, y) isIn (s * t) <=> x isIn s /\ y isIn t

. inl x = (x, False)

. inr x = (x, True);

%% Type Constructors -----------------------------------------------------

Bool : Type

Set : Type -> Type := \ S : Type . S ->? Unit

%% Type Variables --------------------------------------------------------

S : Type %(var_1)%

T : Type %(var_2)%

%% Assumptions -----------------------------------------------------------

False : Bool %(op)%

True : Bool %(op)%

__*__

: forall S : Type; T : Type . Set S * Set T -> Set (S * T) %(op)%

__\\__ : forall S : Type . Set S * Set S -> Set S %(op)%

__disjoint__ : forall S : Type . Pred (Set S * Set S) %(op)%

__disjointUnion__

: forall S : Type . Set S * Set S -> Set (S * Bool) %(op)%

__intersection__ : forall S : Type . Set S * Set S -> Set S %(op)%

__isIn__ : forall S : Type . S * Set S ->? Unit %(op)%

__subset__ : forall S : Type . Pred (Set S * Set S) %(op)%

__union__ : forall S : Type . Set S * Set S -> Set S %(op)%

emptySet : forall S : Type . Set S %(op)%

inl : forall S : Type . S -> S * Bool %(op)%

inr : forall S : Type . S -> S * Bool %(op)%

{__} : forall S : Type . S -> Set S %(op)%

%% Variables -------------------------------------------------------------

s : Set S

s' : Set S

t : Set T

x : S

x' : S

y : T

%% Sentences -------------------------------------------------------------

not x isIn emptySet

x isIn { x' } <=> x = x'

x isIn s <=> s x

s subset s' <=> forall x : S . x isIn s => x isIn s'

x isIn (s union s') <=> x isIn s \/ x isIn s'

x isIn (s intersection s') <=> x isIn s /\ x isIn s'

x isIn (s \\ s') <=> x isIn s /\ not x isIn s'

s disjoint s' <=> s intersection s' = emptySet

(x, y) isIn (s * t) <=> x isIn s /\ y isIn t

inl x = (x, False)

inr x = (x, True)

%% Diagnostics -----------------------------------------------------------

### Hint 1.6, is type variable 'S'

### Hint 1.8, is type variable 'T'

### Hint 3.19, rebound type variable 'S'

### Hint 3.8, redeclared type 'Pred'

### Hint 4.17, rebound type variable 'S'

### Hint 16.11, not a class 'S'

### Hint 16.14, not a class 'S'

### Hint 16.19, not a class 'T'

### Hint 16.24, not a kind 'Set (S)'

### Hint 16.27, not a kind 'Set (S)'

### Hint 16.37, not a kind 'Set (T)'

### Hint 20.32, not a class 'S'

### Hint 20.31, rebound variable 'x'