Lambda calculus

1. Lambda calculus eunmin

2. λ x . x y x y x z function(x) { return x; } y(x); y(x(z));

3. λ x . (λ y . y x) λ x y . y x function(x) { return function(y) { return y(x); }; } function(x, y) { return y(x); }

4. (λ x . x) y (λ x . x) (λ x . x) (function(x) { return x; })(y); (function(x) { return x; })(function(x) { return x; });

5. α-equivalent λ x . x = λ y . y λ x . x != λ x . y (λ x . x) (λ x . x) = (λ x . x) (λ y . y)

6. β-reduction (λ x. x) y y (λ x . x) (λ y . y) λ y . y (λ x y . x y) (λ x . x) (λ x . x) (λ x . x) (λ x . x) λ x . x

7. β-reduction (λ x y . x y) (λ x . x) (λ x . x) (function (x, y) { return x(y); })(function (x) { return x; }, function (x) { return x; });

11. β-reduction (λ x y . x y) (λ x . x) (λ x . x) (λ x . x) (λ x . x) (function (x) { return x; })(function (x) { return x; });

14. β-reduction (λ x y . x y) (λ x . x) (λ x . x) (λ x . x) (λ x . x) λ x . x function (x) { return x; }

15. Church encoding λ s z . z

16. Church encoding λ s z . z 0

17. Church encoding λ s z . z 0 λ s z . s z 1

18. Church encoding λ s z . z 0 λ s z . s z 1 λ s z . s (s z) 2

19. Church encoding λ s z . z 0 λ s z . s z 1 λ s z . s (s z) 2 λ s z . s (s (s z)) 3

20. Church encoding 0

21. Encoding 0 1

22. Encoding 0 1 2

23. Encoding 0 1 2 3

24. Encoding 0 1 2 3 λ s z . z 0 λ s z . s z 1 λ s z . s (s z) 2 λ s z . s (s (s z)) 3

25. + λ x y. (λ s z . (x s (y s z))) function (x, y) { return function (s, z) { return x(s(y(s, z))); } }

26. 1 + 2 + 1 2 (λ x y. (λ s z . (x s (y s z))))(λ s z . s z)(λ s z . s (s z))

31. 1 + 2 + 1 2 (λ x y. (λ s z . (x s (y s z))))(λ s2 z2 . s2 z2)(λ s3 z3 . s3 (s3 z3)) ;; α-equivalent

32. 1 + 2 + 1 2 (λ x y. (λ s z . (x s (y s z))))(λ s2 z2 . s2 z2)(λ s3 z3 . s3 (s3 z3))

33. 1 + 2 + 1 2 (λ x y. (λ s z . (x s (y s z))))(λ s2 z2 . s2 z2)(λ s3 z3 . s3 (s3 z3))

34. 1 + 2 + 1 2 (λ x y. (λ s z . (x s (y s z))))(λ s2 z2 . s2 z2)(λ s3 z3 . s3 (s3 z3)) λ s z . ((λ s2 z2 . s2 z2) s ((λ s3 z3 . s3 (s3 z3)) s z))) ;; β-reduction

35. 1 + 2 + 1 2 (λ x y. (λ s z . (x s (y s z))))(λ s2 z2 . s2 z2)(λ s3 z3 . s3 (s3 z3)) λ s z . (λ s2 z2 . s2 z2) s ((λ s3 z3 . s3 (s3 z3)) s z))

38. 1 + 2 + 1 2 (λ x y. (λ s z . (x s (y s z))))(λ s2 z2 . s2 z2)(λ s3 z3 . s3 (s3 z3)) λ s z . (λ s2 z2 . s2 z2) s ((λ s3 z3 . s3 (s3 z3)) s z)) λ s z . s ((λ s3 z3 . s3 (s3 z3)) s z))

42. 1 + 2 + 1 2 (λ x y. (λ s z . (x s (y s z))))(λ s2 z2 . s2 z2)(λ s3 z3 . s3 (s3 z3)) λ s z . (λ s2 z2 . s2 z2) s ((λ s3 z3 . s3 (s3 z3)) s z)) λ s z . s ((λ s3 z3 . s3 (s3 z3)) s z)) λ s z . s (s (s z))

43. 1 + 2 + 1 2 (λ x y. (λ s z . (x s (y s z))))(λ s2 z2 . s2 z2)(λ s3 z3 . s3 (s3 z3)) λ s z . (λ s2 z2 . s2 z2) s ((λ s3 z3 . s3 (s3 z3)) s z)) λ s z . s ((λ s3 z3 . s3 (s3 z3)) s z)) λ s z . s (s (s z))

44. 1 + 2 + 1 2 (λ x y. (λ s z . (x s (y s z))))(λ s2 z2 . s2 z2)(λ s3 z3 . s3 (s3 z3)) λ s z . (λ s2 z2 . s2 z2) s ((λ s3 z3 . s3 (s3 z3)) s z)) λ s z . s ((λ s3 z3 . s3 (s3 z3)) s z)) λ s z . s (s (s z)) λ s z . z 0 λ s z . s z 1 λ s z . s (s z) 2 λ s z . s (s (s z)) 3

45. 1 + 2 + 1 2 (λ x y. (λ s z . (x s (y s z))))(λ s2 z2 . s2 z2)(λ s3 z3 . s3 (s3 z3)) λ s z . (λ s2 z2 . s2 z2) s ((λ s3 z3 . s3 (s3 z3)) s z)) λ s z . s ((λ s3 z3 . s3 (s3 z3)) s z)) λ s z . s (s (s z)) = 3 λ s z . z 0 λ s z . s z 1 λ s z . s (s z) 2 λ s z . s (s (s z)) 3

46. Boolean λ t f . t true

47. Boolean λ t f . t true λ t f . f false function(t, f) { return t; } function(t, f) { return f; }

48. if-else λ b t f . b t f function(b, t, f) { return b(t, f); }

49. if true 1 else 2 if (true) { return 1; } else { return 2; }

50. if (true) { return 1; } else { return 2; } if-else = λ b t f . b t f true = λ t f . t 1 = λ s z . s z 2 = λ s z . s (s z) if-else true 1 2 (λ b t f . b t f)(λ t f . t)(λ s z . s z)(λ s z . s (s z))

55. if (true) { return 1; } else { return 2; } if-else = λ b t f . b t f true = λ t f . t 1 = λ s z . s z 2 = λ s z . s (s z) if-else true 1 2 (λ b t f . b t f)(λ t f . t)(λ s z . s z)(λ s z . s (s z)) (λ t f . t)(λ s z . s z)(λ s z . s (s z)) ;; β-reduction

56. if (true) { return 1; } else { return 2; } if-else = λ b t f . b t f true = λ t f . t 1 = λ s z . s z 2 = λ s z . s (s z) if-else true 1 2 (λ b t f . b t f)(λ t f . t)(λ s z . s z)(λ s z . s (s z)) (λ t f . t)(λ s z . s z)(λ s z . s (s z))

57. if (true) { return 1; } else { return 2; } if-else = λ b t f . b t f true = λ t f . t 1 = λ s z . s z 2 = λ s z . s (s z) if-else true 1 2 (λ b t f . b t f)(λ t f . t)(λ s z . s z)(λ s z . s (s z)) (λ t f . t)(λ s z . s z)(λ s z . s (s z)) (λ s z . s z) ;; β-reduction

58. if (true) { return 1; } else { return 2; } if-else = λ b t f . b t f true = λ t f . t 1 = λ s z . s z 2 = λ s z . s (s z) if-else true 1 2 (λ b t f . b t f)(λ t f . t)(λ s z . s z)(λ s z . s (s z)) (λ t f . t)(λ s z . s z)(λ s z . s (s z)) λ s z . s z

59. if (true) { return 1; } else { return 2; } if-else = λ b t f . b t f true = λ t f . t 1 = λ s z . s z 2 = λ s z . s (s z) if-else true 1 2 (λ b t f . b t f)(λ t f . t)(λ s z . s z)(λ s z . s (s z)) (λ t f . t)(λ s z . s z)(λ s z . s (s z)) λ s z . s z = 1

60. Loop (Recursion) f = λ x . f x function f(x) { return f(x); }

61. Loop (Recursion) f = λ x . f x function f(x) { return f(x); }

62. Loop (Recursion) Y combinator Y = λ y . (λ x . y (x x)) (λ x . y (x x)) Y f = f Y f = f f Y f = f f f Y f = … loop f with Y Lambda Abstraction

63. Y f (λ y . (λ x . y (x x)) (λ x . y (x x)))f

64. Y f (λ y . (λ x . y (x x)) (λ x . y (x x)))f (λ y . (λ x2 . y (x2 x2)) (λ x3 . y (x3 x3)))f ;; α-equivalent

65. Y f (λ y . (λ x . y (x x)) (λ x . y (x x)))f (λ y . (λ x2 . y (x2 x2)) (λ x3 . y (x3 x3)))f

66. Y f (λ y . (λ x . y (x x)) (λ x . y (x x)))f (λ y . (λ x2 . y (x2 x2)) (λ x3 . y (x3 x3)))f (λ x2 . f (x2 x2)) (λ x3 . f (x3 x3)) ;; β-reduction

67. Y f (λ y . (λ x . y (x x)) (λ x . y (x x)))f (λ y . (λ x2 . y (x2 x2)) (λ x3 . y (x3 x3)))f (λ x2 . f (x2 x2)) (λ x3 . f (x3 x3))

68. Y f (λ y . (λ x . y (x x)) (λ x . y (x x)))f (λ y . (λ x2 . y (x2 x2)) (λ x3 . y (x3 x3)))f (λ x2 . f (x2 x2)) (λ x3 . f (x3 x3))

69. Y f (λ y . (λ x . y (x x)) (λ x . y (x x)))f (λ y . (λ x2 . y (x2 x2)) (λ x3 . y (x3 x3)))f (λ x2 . f (x2 x2)) (λ x3 . f (x3 x3)) f (λ x3 . f (x3 x3)) (λ x3 . f (x3 x3)) ;; β-reduction

70. Y f (λ y . (λ x . y (x x)) (λ x . y (x x)))f (λ y . (λ x2 . y (x2 x2)) (λ x3 . y (x3 x3)))f (λ x2 . f (x2 x2)) (λ x3 . f (x3 x3)) f (λ x3 . f (x3 x3)) (λ x3 . f (x3 x3))

71. Y f (λ y . (λ x . y (x x)) (λ x . y (x x)))f (λ y . (λ x2 . y (x2 x2)) (λ x3 . y (x3 x3)))f (λ x2 . f (x2 x2)) (λ x3 . f (x3 x3)) f (λ x3 . f (x3 x3)) (λ x3 . f (x3 x3)) f (λ x . y (x x)) (λ x . y (x x)) ;; α-equivalent

72. Y f (λ y . (λ x . y (x x)) (λ x . y (x x)))f (λ y . (λ x2 . y (x2 x2)) (λ x3 . y (x3 x3)))f (λ x2 . f (x2 x2)) (λ x3 . f (x3 x3)) f (λ x3 . f (x3 x3)) (λ x3 . f (x3 x3)) f (λ x . y (x x)) (λ x . y (x x))

73. Y f (λ y . (λ x . y (x x)) (λ x . y (x x)))f (λ y . (λ x2 . y (x2 x2)) (λ x3 . y (x3 x3)))f (λ x2 . f (x2 x2)) (λ x3 . f (x3 x3)) f (λ x3 . f (x3 x3)) (λ x3 . f (x3 x3)) f (λ x . f (x x)) (λ x . f (x x))

74. Y f (λ y . (λ x . y (x x)) (λ x . y (x x)))f (λ y . (λ x2 . y (x2 x2)) (λ x3 . y (x3 x3)))f (λ x2 . f (x2 x2)) (λ x3 . f (x3 x3)) f (λ x3 . f (x3 x3)) (λ x3 . f (x3 x3)) f Y f

Lambda calculus

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Lambda calculus