var a : Type; b < a

op twice2 : forall a : Type; b < a : Type . (a ->? b) -> a ->? b

= \ ((var f : a ->? b))((var x : a))

. (var f : a ->? b) ((var f : a ->? b) (var x : a)) as b

op twice3 : forall a : Type; b < a : Type . (a ->? b) * a ->? b

= \ ((var f : a ->? b), (var x : a))

. (var f : a ->? b) ((var f : a ->? b) (var x : a))

as (a ->? b) * a ->? b

op twice : forall a : Type; b < a : Type . (a ->? b) -> a ->? b

= \ ((var f : a ->? a))((var x : a))

. (var f : a ->? a) ((var f : a ->? a) (var x : a))

as (a ->? b) -> a ->? b

var f : a ->? b; x : a

. (fun __=__[b] : forall a : Type . a * a ->? Unit)

((((op twice[a; b] :

forall a : Type; b < a : Type . (a ->? b) -> a ->? b)

(var f : a ->? b))

(var x : a),

(var f : a ->? b) ((var f : a ->? b) (var x : a))))

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

? : +Type -> Type

Logical : Type := ? Unit

Pred : -Type -> Type := \ a : -Type . a ->? Unit

Unit : Type

__*__ : +Type -> +Type -> Type

__*__*__ : +Type -> +Type -> +Type -> Type

__*__*__*__ : +Type -> +Type -> +Type -> +Type -> Type

__*__*__*__*__ : +Type -> +Type -> +Type -> +Type -> +Type -> Type

__-->__ : -Type -> +Type -> Type < (__-->?__, __->__)

__-->?__ : -Type -> +Type -> Type < __->?__

__->__ : -Type -> +Type -> Type < __->?__

__->?__ : -Type -> +Type -> Type

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

a : Type %(var_1)%

b < a : Type %(var_2)%

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

__/\__ : ? Unit * ? Unit ->? Unit %(fun)%

__<=>__ : ? Unit * ? Unit ->? Unit %(fun)%

__=__ : forall a : Type . a * a ->? Unit %(fun)%

__=>__ : ? Unit * ? Unit ->? Unit %(fun)%

__=e=__ : forall a : Type . a * a ->? Unit %(fun)%

__\/__ : ? Unit * ? Unit ->? Unit %(fun)%

__if__ : ? Unit * ? Unit ->? Unit %(fun)%

__when__else__ : forall a : Type . a * ? Unit * a ->? a %(fun)%

bottom : forall a : Type . a %(fun)%

def__ : forall a : Type . a ->? Unit %(fun)%

false : Unit %(fun)%

not__ : ? Unit ->? Unit %(fun)%

true : Unit %(fun)%

twice

: forall a : Type; b < a : Type . (a ->? b) -> a ->? b

%(op)%

= \ ((var f : a ->? a))((var x : a))

. (var f : a ->? a) ((var f : a ->? a) (var x : a))

as (a ->? b) -> a ->? b

twice2

: forall a : Type; b < a : Type . (a ->? b) -> a ->? b

%(op)%

= \ ((var f : a ->? b))((var x : a))

. (var f : a ->? b) ((var f : a ->? b) (var x : a)) as b

twice3

: forall a : Type; b < a : Type . (a ->? b) * a ->? b

%(op)%

= \ ((var f : a ->? b), (var x : a))

. (var f : a ->? b) ((var f : a ->? b) (var x : a))

as (a ->? b) * a ->? b

�__ : ? Unit ->? Unit %(fun)%

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

f : a ->? b

x : a

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

forall f : a ->? b; x : a

. ((twice2 (f)) x) = (f (f (x)) as b) %(def_twice2)%

twice3 =

(\ (f, x) . f (f (x)) as (a ->? b) * a ->? b) %(def_twice3)%

twice =

(\ (f)(x) . f (f (x)) as (a ->? b) -> a ->? b) %(def_twice)%

((twice (f)) x) = (f (f (x)))

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

*** Hint 1.6, is type variable 'a'

*** Hint 9.7, not a kind 'a ->? b'

*** Hint 9.20, not a class 'a'