flatmap15

(slide "start")
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Existential Lambda Calculus
or: Less Unitype, More Unicode

(start-lamb "C-e"
            "C-k"
            "C-S-d")
Standard ML of New Jersey v110.76 [built: Sun Jul 14 09:59:19 2013]
-
repl.repl ();
Not really Standard ML of New Jersey v110.76 [built: kind of]


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(slide "untyped")
------------------------------------------------------------
untyped:

Syntax:
Exp u ::= x         variable
          λx.u      abstraction
          u1 u2     application
 
Computation rule:
(λx.u1) u2
[u2/x]u1

e.g.
#u (λa.λb.λc.a c c) foo bar quux
#u (λb.λc.foo c c) bar quux
#u (λc.foo c c) quux
#u foo quux quux

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(slide "fals0")
------------------------------------------------------------
0 looked like λf.λx.x
  (while 1 looked like λf.λx.f x
         2 looked like λf.λx.f (f x)
         etc)

false looked like λa.λb.b
  (while true looked like λa.λb.a)
#u (λa.λb.b) then else
#u (λb.b) else
#u else
------------------------------------------------------------


(slide "untyped")
------------------------------------------------------------
untyped:

Syntax:
Exp u ::= x         variable
          λ(x : t).u      abstraction
          u1 u2     application
          ΛX.u      type abstraction
 
Computation rule:
(λx.u1) u2
[u2/x]u1

ΛX.λ(x : X).x
ΛX.λ(x : X).x : ΠX.X → X

e.g.
#u (λa.λb.λc.a c c) foo bar quux

------------------------------------------------------------


(slide "typed1")
------------------------------------------------------------
   Jean-Yves Girard's Système F
or John C. Reynolds' polymorphic lambda calculus

Exp u ::= x             variable
          λ(x : t).u1   abstraction
          u1 u2         application
          ΛT.u          type abstraction

Typ t ::= T             variable
          t1 → t2       function
          ΠT.t          type function thing

------------------------------------------------------------

(slide "typed2")
------------------------------------------------------------
the type of
    λ(x : X).meep  is X → the type of meep
                   (assuming that x is of type X in meep)

    ΛX.meep        is ΠX.the type of meep
                   (assuming that X is a type in meep)


computiony:

(λ(x : X).meep) moop     (where moop is of type X)
[moop/x]meep             meep, but replace x with moop


(ΛX.meep) moop           (where moop is a type)
[moop/X]meep             meep, but replace X with moop

------------------------------------------------------------

(slide "kindalike")
------------------------------------------------------------
so,
ΠX.X → X                      (the type of ΛX.λ(x : X).x)
  is a little bit like
     'a -> 'a                 (the type of fn x => x)
  or t -> t                   (the type of \x -> x)
  or (X : Type) -> X -> X     (the type of \X => \x : X => x
                                        or \X, x : X => x)

------------------------------------------------------------


(slide "fals0")
------------------------------------------------------------
0 looked like λf.λx.x
  (while 1 looked like λf.λx.f x
         2 looked like λf.λx.f (f x)
         etc)

false looked like λa.λb.b
  (while true looked like λa.λb.a)

ΛX.λ(f : X → X).λ(x : X).x
ΛX.λ(f : X → X).λ(x : X).x : nat
ΛX.λ(f : X → X).λ(x : X).x : ΠX.(X → X) → X → X
nat ≜ ΠX.(X → X) → X → X
nat type

true ≜ ΛX.λ(a : X).λ(b : X).b
true : ΠX.X → X → X
ΛX.λ(a : X).λ(b : X).b : ΠX.X → X → X
false ≜ ΛX.λ(a : X).λ(b : X).a
false : ΠX.X → X → X
ΛX.λ(a : X).λ(b : X).a : ΠX.X → X → X

------------------------------------------------------------

(slide "nat")
------------------------------------------------------------
0 ≜ ΛX.λ(f : X → X).λ(x : X).x
0 : nat
1 ≜ ΛX.λ(f : X → X).λ(x : X).f x
1 : nat
2 ≜ ΛX.λ(f : X → X).λ(x : X).f (f x)
2 : nat
etc ...

S ≜ λ(n : nat).ΛX.λ(f : X → X).λ(x : X).f (n X f x)
S : nat → nat
+ ≜ λ(a : nat).λ(b : nat).a nat S b
+ : nat → nat → nat

S (S 2)

#u λf.λx.f (f (f (f x)))

+ 2 1
(λ(b : ΠX.(X → X) → X → X).(ΛX.λ(f : X → X).λ(x : X).f (f x)) (ΠX.(X → X) → X → X) (λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) b) (ΛX.λ(f : X → X).λ(x : X).f x) : nat
(ΛX.λ(f : X → X).λ(x : X).f (f x)) (ΠX.(X → X) → X → X) (λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) (ΛX.λ(f : X → X).λ(x : X).f x) : nat
(λ(f : (ΠX.(X → X) → X → X) → ΠX.(X → X) → X → X).λ(x : ΠX.(X → X) → X → X).f (f x)) (λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) (ΛX.λ(f : X → X).λ(x : X).f x) : nat
(λ(x : ΠX.(X → X) → X → X).(λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) ((λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) x)) (ΛX.λ(f : X → X).λ(x : X).f x) : nat
(λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) ((λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) (ΛX.λ(f : X → X).λ(x : X).f x)) : nat
ΛX.λ(f : X → X).λ(x : X).f ((λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) (ΛX.λ(f : X → X).λ(x : X).f x) X f x) : nat
ΛX.λ(f : X → X).λ(x : X).f ((ΛX.λ(f : X → X).λ(x : X).f ((ΛX.λ(f : X → X).λ(x : X).f x) X f x)) X f x) : nat
ΛX.λ(f : X → X).λ(x : X).f ((λ(f : X → X).λ(x : X).f ((ΛX.λ(f : X → X).λ(x : X).f x) X f x)) f x) : nat
ΛX.λ(f : X → X).λ(x : X).f ((λ(x : X).f ((ΛX.λ(f : X → X).λ(x : X).f x) X f x)) x) : nat
ΛX.λ(f : X → X).λ(x : X).f (f ((ΛX.λ(f : X → X).λ(x : X).f x) X f x)) : nat
ΛX.λ(f : X → X).λ(x : X).f (f ((λ(f : X → X).λ(x : X).f x) f x)) : nat
ΛX.λ(f : X → X).λ(x : X).f (f ((λ(x : X).f x) x)) : nat
ΛX.λ(f : X → X).λ(x : X).f (f (f x)) : nat

+ 2 2
(λ(b : ΠX.(X → X) → X → X).(ΛX.λ(f : X → X).λ(x : X).f (f x)) (ΠX.(X → X) → X → X) (λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) b) (ΛX.λ(f : X → X).λ(x : X).f (f x)) : nat
(ΛX.λ(f : X → X).λ(x : X).f (f x)) (ΠX.(X → X) → X → X) (λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) (ΛX.λ(f : X → X).λ(x : X).f (f x)) : nat
(λ(f : (ΠX.(X → X) → X → X) → ΠX.(X → X) → X → X).λ(x : ΠX.(X → X) → X → X).f (f x)) (λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) (ΛX.λ(f : X → X).λ(x : X).f (f x)) : nat
(λ(x : ΠX.(X → X) → X → X).(λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) ((λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) x)) (ΛX.λ(f : X → X).λ(x : X).f (f x)) : nat
(λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) ((λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) (ΛX.λ(f : X → X).λ(x : X).f (f x))) : nat
ΛX.λ(f : X → X).λ(x : X).f ((λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) (ΛX.λ(f : X → X).λ(x : X).f (f x)) X f x) : nat
ΛX.λ(f : X → X).λ(x : X).f ((ΛX.λ(f : X → X).λ(x : X).f ((ΛX.λ(f : X → X).λ(x : X).f (f x)) X f x)) X f x) : nat
ΛX.λ(f : X → X).λ(x : X).f ((λ(f : X → X).λ(x : X).f ((ΛX.λ(f : X → X).λ(x : X).f (f x)) X f x)) f x) : nat
ΛX.λ(f : X → X).λ(x : X).f ((λ(x : X).f ((ΛX.λ(f : X → X).λ(x : X).f (f x)) X f x)) x) : nat
ΛX.λ(f : X → X).λ(x : X).f (f ((ΛX.λ(f : X → X).λ(x : X).f (f x)) X f x)) : nat
ΛX.λ(f : X → X).λ(x : X).f (f ((λ(f : X → X).λ(x : X).f (f x)) f x)) : nat
ΛX.λ(f : X → X).λ(x : X).f (f ((λ(x : X).f (f x)) x)) : nat
ΛX.λ(f : X → X).λ(x : X).f (f (f (f x))) : nat


+ 2 false
typefail :(

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(slide "sumrules")
------------------------------------------------------------
cow product
sum rules:

+, formation
  tl type    tr type
 --------------------
    tl + tr type

+, intro
       u : tl                  u : tr
 ------------------     -------------------
  left u : tl + tr       right u : tl + tr

+, elim
  u : tl + tr    xl : tl ⊢ ul : t    xr : tr ⊢ ur : t
 -----------------------------------------------------
      case(u, left(xl) => ul, right(xr) => ur) : t

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(slide "summl")
------------------------------------------------------------
datatype ('a, 'b) either = Left of 'a
                         | Right of 'b
case (Right ()) of
    Left x  => x + x
  | Right _ => 0

case (Left 2) of
    Left x  => x + x
  | Right _ => 0

------------------------------------------------------------

left ≜ ΛL.ΛR.λ(l : L).ΛT.λ(lf : L → T).λ(rf : R → T).lf l
left : ΠL.ΠR.L → ΠT.(L → T) → (R → T) → T
right ≜ ΛL.ΛR.λ(r : R).ΛT.λ(lf : L → T).λ(rf : R → T).rf r
right : ΠL.ΠR.R → ΠT.(L → T) → (R → T) → T

unit ≜ ΠX.X → X
unit type
u ≜ ΛX.λ(x : X).x
u : unit

some ≜ ΛA.λ(a : A).left A unit a
some : ΠA.A → ΠT.(A → T) → (unit → T) → T

none ≜ ΛA.right A unit u
none : ΠA.ΠT.(A → T) → (unit → T) → T

natoption ≜ ΠT.(nat → T) → (unit → T) → T
natoption type

nonat ≜ none nat
nonat : natoption

somenat ≜ λ(n : nat).some nat n
somenat : nat → natoption

nonat nat (λ(n : nat).+ n n) (λ(_ : unit).0)
(ΛL.ΛR.λ(r : R).ΛT.λ(lf : L → T).λ(rf : R → T).rf r) (ΠX.(X → X) → X → X) (ΠX.X → X) (ΛX.λ(x : X).x) (ΠX.(X → X) → X → X) (λ(n : ΠX.(X → X) → X → X).(λ(a : ΠX.(X → X) → X → X).λ(b : ΠX.(X → X) → X → X).a (ΠX.(X → X) → X → X) (λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) b) n n) (λ(_ : ΠX.X → X).ΛX.λ(f : X → X).λ(x : X).x) : nat
(ΛR.λ(r : R).ΛT.λ(lf : (ΠX.(X → X) → X → X) → T).λ(rf : R → T).rf r) (ΠX.X → X) (ΛX.λ(x : X).x) (ΠX.(X → X) → X → X) (λ(n : ΠX.(X → X) → X → X).(λ(a : ΠX.(X → X) → X → X).λ(b : ΠX.(X → X) → X → X).a (ΠX.(X → X) → X → X) (λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) b) n n) (λ(_ : ΠX.X → X).ΛX.λ(f : X → X).λ(x : X).x) : nat
(λ(r : ΠX.X → X).ΛT.λ(lf : (ΠX.(X → X) → X → X) → T).λ(rf : (ΠX.X → X) → T).rf r) (ΛX.λ(x : X).x) (ΠX.(X → X) → X → X) (λ(n : ΠX.(X → X) → X → X).(λ(a : ΠX.(X → X) → X → X).λ(b : ΠX.(X → X) → X → X).a (ΠX.(X → X) → X → X) (λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) b) n n) (λ(_ : ΠX.X → X).ΛX.λ(f : X → X).λ(x : X).x) : nat
(ΛT.λ(lf : (ΠX.(X → X) → X → X) → T).λ(rf : (ΠX.X → X) → T).rf (ΛX.λ(x : X).x)) (ΠX.(X → X) → X → X) (λ(n : ΠX.(X → X) → X → X).(λ(a : ΠX.(X → X) → X → X).λ(b : ΠX.(X → X) → X → X).a (ΠX.(X → X) → X → X) (λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) b) n n) (λ(_ : ΠX.X → X).ΛX.λ(f : X → X).λ(x : X).x) : nat
(λ(lf : (ΠX.(X → X) → X → X) → ΠX.(X → X) → X → X).λ(rf : (ΠX.X → X) → ΠX.(X → X) → X → X).rf (ΛX.λ(x : X).x)) (λ(n : ΠX.(X → X) → X → X).(λ(a : ΠX.(X → X) → X → X).λ(b : ΠX.(X → X) → X → X).a (ΠX.(X → X) → X → X) (λ(n : ΠX.(X → X) → X → X).ΛX.λ(f : X → X).λ(x : X).f (n X f x)) b) n n) (λ(_ : ΠX.X → X).ΛX.λ(f : X → X).λ(x : X).x) : nat
(λ(rf : (ΠX.X → X) → ΠX.(X → X) → X → X).rf (ΛX.λ(x : X).x)) (λ(_ : ΠX.X → X).ΛX.λ(f : X → X).λ(x : X).x) : nat
(λ(_ : ΠX.X → X).ΛX.λ(f : X → X).λ(x : X).x) (ΛX.λ(x : X).x) : nat
ΛX.λ(f : X → X).λ(x : X).x : nat










(slide "y")
------------------------------------------------------------
#u λf.(λx.f (x x)) (λx.f (x x))

#plox ΛX.λ(f : X).(λ(x : X).f (x x)) (λ(x : X).f (x x))
ΛX.λ(f : X).f ((λ(x : X).f (x x)) (λ(x : X).f (x x))) : typefail :(
ΛX.λ(f : X).f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))) : typefail :(
ΛX.λ(f : X).f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))) : typefail :(
ΛX.λ(f : X).f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
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ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
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ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
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ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
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ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
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ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
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ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
ΛX.λ(f : X).f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f ((λ(x : X).f (x x)) (λ(x : X).f (x x)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) : typefail :(
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