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Lambda as JS, or A Flock of Functions: Combinators, Lambda Calculus, and Church Encodings in JavaScript

Lambda as JS, or A Flock of Functions: Combinators, Lambda Calculus, and Church Encodings in JavaScript

🔗 Repo: https://github.com/glebec/lambda-talk
🔗 Video Part I: https://youtu.be/3VQ382QG-y4
🔗 Video Part II: https://youtu.be/pAnLQ9jwN-E

A presentation given at Fullstack Academy, by instructor Gabriel Lebec. The Lambda Calculus is a symbol manipulation system which suffices to calculate anything calculable. Developed in the 1930s by Alonzo Church, this branch of pure math forms the basis of many functional programming languages (such as Scheme and Haskell), and is explained here through the familiar lens of JavaScript. Along the way, related concepts like combinatory logic (investigated by Haskell B. Curry) and encodings are examined, building up boolean logic and arithmetic from scratch.

(Errata: 307–311 & 313, the JS function `eq` should be written as `gt`.)

Gabriel Lebec

August 24, 2017
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Transcript

  1. as.js
    A FL O C K of FU N C T I O N S
    COMBINATORS, LAMBDA CALCULUS,
    & CHURCH ENCODINGS in JAVASCRIPT

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  2. glebec
    glebec
    glebec
    glebec
    g_lebec

    Gabriel Lebec
    github.com/glebec/lambda-talk

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  3. a.a
    IDENTITY

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  4. λ
    JS I = a => a
    I := a.a

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  5. λ
    JS I = a => a
    I := a.a

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  6. λ
    JS I = a => a
    I := a.a

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  7. λ
    JS I = a => a
    I := a.a

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  8. λ
    JS I = a => a
    I := a.a

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  9. λ
    JS I(x) === ?
    I x = ?
    I := a.a
    I = a => a

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  10. λ
    JS I(x) === x
    I x = x
    I := a.a
    I = a => a

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  11. λ
    JS I(I) === ?
    I I = ?
    I := a.a
    I = a => a

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  12. λ
    JS I(I) === I
    I I = I
    I := a.a
    I = a => a

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  13. id 5 == 5

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  14. ?

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  15. a.a
    FUNCTION
    SIGNIFIER

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  16. a.a
    FUNCTION
    SIGNIFIER
    PARAMETER VARIABLE

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  17. a.a
    FUNCTION
    SIGNIFIER
    PARAMETER VARIABLE
    RETURN
    EXPRESSION

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  18. a.a
    FUNCTION
    SIGNIFIER
    PARAMETER VARIABLE
    RETURN
    EXPRESSION
    LAMBDA ABSTRACTION

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  19. -CALCULUS SYNTAX
    expression ::= variable identifier
    | expression expression application
    | variable . expression abstraction
    | ( expression ) grouping

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  20. λ JS

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  21. VARIABLES
    x x
    (a) (a)

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  22. f a f(a)
    f a b f(a)(b)
    (f a) b (f(a))(b)
    f (a b) f(a(b))
    APPLICATIONS

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  23. a.b a => b
    a.b x a => b(x)
    a.(b x) a => (b(x))
    (a.b) x (a => b)(x)
    a.b.a a => b => a
    a.(b.a) a => (b => a)
    ABSTRACTIONS

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  24. ((a.a)b.c.b)(x)e.f
    β-REDUCTION

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  25. ((a.a)b.c.b)(x)e.f
    β-REDUCTION

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  26. ((a.a)b.c.b)(x)e.f
    β-REDUCTION

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  27. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    β-REDUCTION

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  28. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    β-REDUCTION

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  29. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    β-REDUCTION

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  30. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION

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  31. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION

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  32. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION

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  33. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    = x
    β-REDUCTION

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  34. ((a.a)b.c.b)(x)e.f
    = (b.c.b) (x)e.f
    = (c.x) e.f
    = x
    β-REDUCTION*
    β-NORMAL FORM
    *not covered: evaluation order, variable collision avoidance

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  35. f.ff
    MOCKINGBIRD

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  36. λ
    JS M = f => f(f)
    M := f.ff

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  37. λ
    JS M = f => f(f)
    M := f.ff

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  38. λ
    JS M = f => f(f)
    M := f.ff

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  39. λ
    JS M = f => f(f)
    M := f.ff

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  40. λ
    JS M(I) === ?
    M I = ?
    M := f.ff

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  41. λ
    JS M(I) === I(I)
    && I(I) === ?
    M I = I I = ?
    M := f.ff

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  42. λ
    JS M(I) === I(I)
    && I(I) === I
    M I = I I = I
    M := f.ff

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  43. λ
    JS M(M) === ?
    M M = ?
    M := f.ff

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  44. λ
    JS M(M) === M(M) === ?
    M M = M M = ?
    M := f.ff

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  45. λ
    JS
    M M = M M = M M = Ω
    // stack overflow
    M := f.ff
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M

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  46. λ
    JS
    ω ω = ω ω = ω ω = Ω
    // stack overflow
    M := f.ff
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M
    M(M) === M(M) === M(M) === M(M) === M

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  47. a.b.c.b a => b => c => b
    abc.b a => b => c => b
    (a, b, c) => b
    =
    ABSTRACTIONS, again

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  48. ((a.a)bc.b)(x)e.f
    β-REDUCTION, again

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  49. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    β-REDUCTION, again

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  50. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    β-REDUCTION, again

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  51. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION, again

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  52. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    = (c.x) e.f
    β-REDUCTION, again

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  53. ((a.a)bc.b)(x)e.f
    = (bc.b) (x)e.f
    = (c.x) e.f
    = x
    β-REDUCTION, again
    β-NORMAL FORM

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  54. ab.a
    KESTREL

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  55. λ
    JS K = a => b => a
    K := ab.a
    = a.b.a

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  56. λ
    JS K = a => b => a
    K := ab.a
    = a.b.a

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  57. λ
    JS K = a => b => a
    K := ab.a
    = a.b.a

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  58. λ
    JS K = a => b => a
    K := ab.a
    = a.b.a

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  59. λ
    JS K(M)(I) === ?
    K M I = ?
    K := ab.a
    K = a => b => a

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  60. λ
    JS K(M)(I) === M
    K M I = M
    K := ab.a
    K = a => b => a

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  61. λ
    JS K(M)(I) === M
    K(I)(M) === ?
    K M I = M
    K I M = ?
    K := ab.a
    K = a => b => a

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  62. λ
    JS K(M)(I) === M
    K(I)(M) === I
    K M I = M
    K I M = I
    K := ab.a
    K = a => b => a

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  63. const 7 2 == 7

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  64. λ
    JS K(I)(x) === I
    K I x = I
    K := ab.a
    K = a => b => a

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  65. λ
    JS K(I)(x)(y) === I(y)
    && I(y) === ?
    K I x y = I y = ?
    K := ab.a
    K = a => b => a

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  66. λ
    JS
    K I x y = I y = y
    K := ab.a
    K = a => b => a
    K(I)(x)(y) === I(y)
    && I(y) === y

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  67. λ
    JS
    K I x y = I y = y
    K := ab.a
    K = a => b => a
    K(I)(x)(y) === I(y)
    && I(y) === y

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  68. λ
    JS
    K I x y = I y = y
    K := ab.a
    K = a => b => a
    K(I)(x)(y) === I(y)
    && I(y) === y

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  69. ab.b
    KITE

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  70. λ
    JS KI = a => b => b
    KI = K(I)
    KI := ab.b
    = K I

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  71. λ
    JS KI(M)(K) === ?
    KI M K = ?
    KI := ab.b
    KI = a => b => b

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  72. λ
    JS KI(M)(K) === K
    KI M K = K
    KI := ab.b
    KI = a => b => b

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  73. λ
    JS KI(M)(K) === K
    KI(K)(M) === ?
    KI M K = K
    KI K M = ?
    KI := ab.b
    KI = a => b => b

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  74. λ
    JS KI(M)(K) === K
    KI(K)(M) === M
    KI M K = K
    KI K M = M
    KI := ab.b
    KI = a => b => b

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  75. ?

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  76. SCHÖNFINKEL CURRY SMULLYAN
    Identitätsfunktion
    Konstante Funktion
    verSchmelzungsfunktion
    verTauschungsfunktion
    Zusammensetzungsf.
    I

    K

    S

    C

    B
    Idiot

    Kestrel

    Starling

    Cardinal

    Bluebird
    Ibis?

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  77. View Slide

  78. ?

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  79. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    TH E FO R M A L I Z AT I O N O F
    MAT H E M AT I C A L LO G I C

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  80. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    FO R M A L NO TAT I O N FO R FU N C T I O N S
    1889
    PE A N O AR I T H M E T I C

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  81. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    AX I O M AT I C LO G I C · FN NO TAT I O N
    FU N C T I O N S A S GR A P H S · CU R RY I N G
    1891

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  82. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    PR I N C I P I A MAT H E M AT I C A
    1910
    RU S S E L L ’S PA R A D OX · FN NO TAT I O N

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  83. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    CO M B I N AT O RY LO G I C
    CO M B I N AT O R S · CU R RY I N G
    1920

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  84. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    FU N C T I O N A L SY S T E M O F SE T TH E O RY
    1925
    (OV E R L A P P E D W I T H CO M B I N AT O RY LO G I C )

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  85. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    CO M B I N AT O RY LO G I C (AG A I N )
    CO M B I N AT O R S · M A N Y C O N T R I B U T I O N S
    1926

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  86. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    D I S C OV E R S SC H Ö N F I N K E L
    “This paper anticipates much of what I have done.”
    1927

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  87. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    IN C O M P L E T E N E S S TH E O R E M S
    1931
    EN D I N G T H E SE A RC H FO R SU F F I C I E N T AX I O M S

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  88. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    -CA L C U L U S
    AN EF F E C T I V E MO D E L O F CO M P U TAT I O N
    1932

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  89. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    I N C O N S I S T E N C Y O F S P E C I A L I Z E D
    1931–1936
    C O N S I S T E N C Y O F P U R E

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  90. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    SO LV E S T H E DE C I S I O N PRO B L E M
    V I A T H E -CA L C U L U S
    1936

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  91. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    SO LV E S T H E DE C I S I O N PRO B L E M
    1936
    V I A T H E TU R I N G MAC H I N E

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  92. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    ES TA B L I S H E S T H E CH U RC H -TU R I N G TH E S I S
    1936
    -CA L C U L U S 㱻 TU R I N G MAC H I N E

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  93. PEANO
    FREGE
    RUSSELL
    SCHÖNFINKEL
    VON NEUMANN
    CURRY CHURCH
    GÖDEL
    TURING
    KLEENE ROSSER
    O B TA I N S PH D U N D E R CH U RC H
    1936–1938
    PU B L I S H E S 1S T FI X E D -PO I N T CO M B I N AT O R

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  94. COMBINATORS
    functions with no free variables
    b.b combinator
    b.a not a combinator
    ab.a combinator
    a.ab not a combinator
    abc.c(e.b) combinator

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  95. COMBINATORS
    Sym. Bird -Calculus Use Haskell
    I Idiot a.a identity id
    M Mockingbird f.ff self-application (cannot define)
    K Kestrel ab.a first, const const
    KI Kite ab.b = KI second const id
    C Cardinal fab.fba reverse arguments flip
    B Bluebird fga.f(ga) 1°-1° composition (.)
    B1
    Blackbird fgab.f(gab) = BBB 1°-2° composition (.) . (.)
    Th
    Thrush af.fa = CI hold an argument flip id
    V Vireo abf.fab = BCT hold a pair of args flip . flip id

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  96. CARDINAL
    fab.fba

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  97. λ
    JS C = f => a => b => f(b)(a)
    C := fab.fba

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  98. λ
    JS C(K)(I)(M) === ?
    C K I M = ?
    C := fab.fba
    C = f => a => b => f(b)(a)

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  99. λ
    JS C(K)(I)(M) === M
    C K I M = M
    C := fab.fba
    C = f => a => b => f(b)(a)

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  100. λ
    JS C(K)(I)(M) === M
    C K I M = M
    C := fab.fba
    C = f => a => b => f(b)(a)

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  101. λ
    JS KI(I)(M) === M
    KI I M = M
    C := fab.fba
    C = f => a => b => f(b)(a)

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  102. COMBINATORS
    Sym. Bird -Calculus Use Haskell
    I Idiot a.a identity id
    M Mockingbird f.ff self-application (cannot define)
    K Kestrel ab.a first, const const
    KI Kite ab.b = KI second const id
    C Cardinal fab.fba reverse arguments flip
    B Bluebird fga.f(ga) 1°-1° composition (.)
    B1
    Blackbird fgab.f(gab) = BBB 1°-2° composition (.) . (.)
    Th
    Thrush af.fa = CI hold an argument flip id
    V Vireo abf.fab = BCT hold a pair of args flip . flip id

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  103. COMBINATORS
    Sym. Bird -Calculus Use Haskell
    I Idiot a.a identity id
    M Mockingbird f.ff self-application (cannot define)
    K Kestrel ab.a first, const const
    KI Kite ab.b = KI = CK second const id
    C Cardinal fab.fba reverse arguments flip
    B Bluebird fga.f(ga) 1°-1° composition (.)
    B1
    Blackbird fgab.f(gab) = BBB 1°-2° composition (.) . (.)
    Th
    Thrush af.fa = CI hold an argument flip id
    V Vireo abf.fab = BCT hold a pair of args flip . flip id

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  104. flip const 1 8 == 8

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  105. -CALCULUS
    abstract symbol rewriting
    functional computation
    TURING MACHINE
    hypothetical device
    state-based computation
    (f.ff)a.a
    purely functional programming languages
    higher-level machine-centric languages
    assembly languages
    machine code
    higher-level abstract stateful languages
    real computers

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  106. TM

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  107. EVERYTHING
    CAN BE
    FUNCTIONS

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  108. TRUE FALSE NOT AND OR BEQ

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  109. λ
    JS const result = bool ? exp1 : exp2

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  110. λ
    JS const result = bool ? exp1 : exp2
    // true

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  111. λ
    JS const result = bool ? exp1 : exp2
    // false

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  112. λ
    JS const result = bool ? exp1 : exp2
    result := ?

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  113. λ
    JS const result = bool ? exp1 : exp2
    result := bool ? exp1 : exp2

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  114. λ
    JS const result = bool ? exp1 : exp2
    result := bool exp1 exp2

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  115. λ
    JS const result = bool (exp1) (exp2)
    result := func exp1 exp2

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  116. λ
    JS
    result := func exp1 exp2
    const result = bool (exp1) (exp2)
    // true

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  117. λ
    JS
    result := func exp1 exp2
    const result = bool (exp1) (exp2)
    // false

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  118. λ
    JS const result = bool (exp1) (exp2)
    result := func exp1 exp2
    TRUE
    FALSE

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  119. λ
    JS const result = bool (exp1) (exp2)
    result := func exp1 exp2
    K
    KI

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  120. λ
    JS const T = K
    const F = KI
    TRUE := K
    FALSE := KI = C K
    CHURCH ENCODINGS: BOOLEANS

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  121. λ
    JS !p
    ! p

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  122. λ
    JS not(p)
    NOT p

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  123. λ
    JS not(p)
    NOT p
    F
    T
    F
    T

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  124. λ
    JS not(T)
    NOT T
    F
    T
    F
    T

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  125. λ
    JS not(F)
    NOT F
    F
    T
    F
    T

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  126. λ
    JS p F T
    F T
    p
    ( )
    ( )

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  127. λ
    JS ( ) ( )
    T F T
    F T
    T
    K
    K
    KI
    KI

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  128. λ
    JS ( ) ( )
    F F T
    F T
    F
    KI
    KI K
    K

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  129. λ
    JS const not = p => p(F)(T)
    NOT := p.pFT

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  130. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq disjunction
    BEQ pq.p q (NOT q) equality

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  131. λ
    JS not(T) === F
    not(F) === T
    NOT T = F
    NOT F = T

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  132. λ
    JS not(K) === KI
    not(KI) === K
    NOT K = KI
    NOT (KI) = K
    ab.a ba.a
    ba.a ab.a

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  133. λ
    JS C(K) (chirp)(tweet) === tweet
    C(KI)(chirp)(tweet) === chirp
    C K = KI
    C (KI) = K

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  134. λ
    JS C(T) (chirp)(tweet) === tweet
    C(F) (chirp)(tweet) === chirp
    C T = F
    C F = T

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  135. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq disjunction
    BEQ pq.p q (NOT q) equality

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  136. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT or C negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq disjunction
    BEQ pq.p q (NOT q) equality

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  137. λ
    JS const and = ?
    AND := ?

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  138. λ
    JS const and = p => q => ?
    AND := pq.?

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  139. λ
    JS const and = p => q => p(?)(¿)
    AND := pq.p?¿

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  140. λ
    JS const and = p => q => p(?)(¿)
    AND := pq.p?¿
    F
    F

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  141. λ
    JS const and = p => q => p(?)(F)
    AND := pq.p?F

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  142. λ
    JS const and = p => q => p(?)(F)
    AND := pq.p?F
    T
    T

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  143. λ
    JS const and = p => q => p(q)(F)
    AND := pq.pqF

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  144. pq.p F
    q

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  145. pq.p
    F
    q

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  146. pq.p
    p
    q

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  147. pq.p p
    q

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  148. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT or C negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq disjunction
    BEQ pq.p q (NOT q) equality

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  149. λ
    JS const or = ?
    OR := ?

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  150. λ
    JS const or = p => q => …
    OR := pq.…

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  151. λ
    JS const or = p => q => p(?)(¿)
    OR := pq.p?¿

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  152. λ
    JS const or = p => q => p(T)(¿)
    OR := pq.pT¿

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  153. λ
    JS const or = p => q => p(T)(q)
    OR := pq.pTq

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  154. λ
    JS const or = p => q => p(p)(q)
    OR := pq.ppq

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  155. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT or C negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq disjunction
    BEQ pq.p q (NOT q) equality

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  156. pq.ppq

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  157. ( pq.ppq ) xy = ?

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  158. ( pq.ppq ) xy = xxy

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  159. ( pq.ppq ) xy = xxy
    ( ? ) xy = xxy

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  160. ( pq.ppq ) xy = xxy
    ( ? ) xy = xxy

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  161. ( pq.ppq ) xy = xxy
    M xy = xxy

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  162. ( pq.ppq ) xy = xxy
    M xy = xxy

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  163. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT or C negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq disjunction
    BEQ pq.p q (NOT q) equality

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  164. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT or C negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq = M* disjunction
    BEQ pq.p q (NOT q) equality

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  165. ( )
    pq.p( ) T
    T F F
    q
    q
    p => q => p(q(T)(F))(q(F)(T))

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  166. ( )
    pq.p
    ( )
    T
    T F
    F
    q
    q

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  167. ( )
    pq.p
    ( )
    T
    T
    F
    F
    q
    q

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  168. ( )
    pq.p
    ( )
    T
    T
    F
    F
    q
    q

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  169. ( )
    pq.p
    ( )
    T
    T
    F
    F
    q
    q

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  170. ( )
    pq.p
    ( )
    T
    T
    F
    F
    q
    q

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  171. ( )
    pq.p
    ( )
    T
    T
    F
    F
    q
    q

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  172. ( )
    pq.p
    ( )
    T
    T
    F
    F
    q
    q
    BOOLEAN EQUALITY

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  173. pq.p
    ( )
    T
    F
    q
    q

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  174. ( )
    pq.p
    q
    NOT
    q

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  175. ( )
    pq.p q
    q NOT
    p => q => p(q)(not(q))

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  176. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT or C negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq = M* disjunction
    BEQ pq.p q (NOT q) equality

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  177. (ONE OF) DE MORGAN'S LAWS
    ¬(P ∧ Q) = (¬P) ∨ (¬Q)
    BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    !(p && q) === (!p) || (!q)

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  178. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  179. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  180. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  181. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  182. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  183. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  184. BEQ (NOT (AND p q)) (OR (NOT p) (NOT q))
    (xy.x y ((fab.fba) y))

    ((fab.fba) ((xy.xyx) p q))

    ((f.ff) ((fab.fba) p) ((fab.fba) q))

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  185. ZERO ONE TWO THREE SUCC

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  186. ZERO ONCE TWICE THRICE SUCC

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  187. λ
    JS n1 = f => a => f(a)
    n2 = f => a => f(f(a))
    n3 = f => a => f(f(f(a)))
    N1 := fa.fa
    N2 := fa.f(fa)
    N3 := fa.f(f(fa))

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  188. λ
    JS n1(not)(T) = not(T) = ?
    N1 NOT T = NOT T = ?
    N1 := fa.fa
    N1 = f => a => f(a)

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  189. λ
    JS n1(not)(T) = not(T) = F
    N1 NOT T = NOT T = F
    N1 := fa.fa
    N1 = f => a => f(a)

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  190. λ
    JS n2(not)(T) = not(not(T)) = ?
    N2 NOT T = NOT (NOT T) = ?
    N2 := fa.f(fa)
    N2 = f => a => f(f(a))

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  191. λ
    JS n2(not)(T) = not(not(T)) = T
    N2 NOT T = NOT (NOT T) = T
    N2 := fa.f(fa)
    N2 = f => a => f(f(a))

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  192. λ
    JS n3(not)(T) = not(not(not(T))) = F
    N3 NOT T =
    NOT (NOT (NOT T))) = F
    N3 := fa.f(f(fa))
    N3 = f => a => f(f(f(a)))

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  193. λ
    JS n0 = f => a => a
    n1 = f => a => f(a)
    n2 = f => a => f(f(a))
    n3 = f => a => f(f(f(a)))
    N0 := fa.a
    N1 := fa.fa
    N2 := fa.f(fa)
    N3 := fa.f(f(fa))

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  194. λ
    JS n0(not)(T) = ?
    N0 NOT T = ?
    N0 := fa.a
    N3 = f => a => a

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  195. λ
    JS n0(not)(T) = T
    N0 NOT T = T
    N0 := fa.a
    N3 = f => a => a

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  196. CHURCH ENCODINGS: NUMERALS
    Sym. Name -Calculus Use
    N0 ZERO fa.a = F apply f no times to a
    N1 ONCE fa.f a = I* apply f once to a
    N2 TWICE fa.f (f a) apply 2-fold f to a
    N3 THRICE fa.f (f (f a)) apply 3-fold f to a
    N4 FOURFOLD fa.f (f (f (f a))) apply 4-fold f to a
    N5 FIVEFOLD fa.f (f (f (f (f a))))) apply 5-fold f to a

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  197. + 1 ?

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  198. PEANO NUMBERS
    SUCC N1 = N2
    SUCC N2 = N3
    SUCC (SUCC N1) = N3

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  199. λ
    JS succ = n => ?
    SUCC := n.?
    SUCC N1 = N2

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  200. λ
    JS succ = n => ?
    SUCC := n.?
    SUCC fa.fa = fa.f(fa)

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  201. λ
    JS succ = n => f => a => f(n(f)(a))
    SUCC := nfa.f(nfa)
    SUCC fa.fa = fa.f(fa)

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  202. λ
    JS succ = n => f => a => f(n(f)(a))
    SUCC := nfa.f(nfa)
    SUCC fa.fa = fa.f(fa)

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  203. λ
    JS succ = n => f => a => f(n(f)(a))
    SUCC := nfa.f(nfa)
    SUCC fa.fa = fa.f(fa)

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  204. SUCC N2 = (nfa.f(nfa)) N2
    = fa.f(N2 f a)
    = fa.f(f(f a)
    = N3

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  205. CHURCH ARITHMETIC
    Name -Calculus Use
    SUCC nfa.f(nfa) successor of n
    ADD nk.n SUCC k = nkf.B (n f) (k f) addition of n and k
    MULT nkf.n(kf) = B multiplication of n and k
    POW nk.kn = Th
    raise n to the power of k
    PRED n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0 predecessor of n
    PRED n.FST (n Φ (PAIR N0 N0)) predecessor of n (easier)
    SUB nk.k PRED n subtract k from n

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  206. BLUEBIRD
    fga.f(ga)

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  207. λ
    JS B = f => g => a => f(g(a))
    B := fga.f(ga)

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  208. λ
    JS B(not)(not)(T) === ?
    B NOT NOT T = ?
    B := fga.f(ga)
    B = f => g => a => f(g(a))

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  209. λ
    JS B(not)(not)(T) === T
    B NOT NOT T = T
    B := fga.f(ga)
    B = f => g => a => f(g(a))

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  210. odd = not . even

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  211. COMBINATORS
    Sym. Bird -Calculus Use Haskell
    I Idiot a.a identity id
    M Mockingbird f.ff self-application (cannot define)
    K Kestrel ab.a true, first, const const
    KI Kite ab.b = KI = CK false, second const id
    C Cardinal fab.fba reverse arguments flip
    B Bluebird fga.f(ga) 1°←1° composition (.)
    B1
    Blackbird fgab.f(gab) = BBB 1°-2° composition (.) . (.)
    Th
    Thrush af.fa = CI hold an argument flip id
    V Vireo abf.fab = BCT hold a pair of args flip . flip id

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  212. λ
    JS succ = n => f => a => f(n(f)(a))
    SUCC := nfa.f(nfa)

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  213. λ
    JS succ = n => f => B(f)(n(f))
    SUCC := nf.Bf(nf)

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  214. CHURCH ARITHMETIC
    Name -Calculus Use
    SUCC nf.B f (nf) = nfa.f(nfa) successor of n
    ADD nk.n SUCC k = nkf.B (n f) (k f) addition of n and k
    MULT nkf.n(kf) = B multiplication of n and k
    POW nk.kn = Th
    raise n to the power of k
    PRED n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0 predecessor of n
    PRED n.FST (n Φ (PAIR N0 N0)) predecessor of n (easier)
    SUB nk.k PRED n subtract k from n

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  215. ADD MULT POW

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  216. λ
    JS add = n => k => ?
    ADD := nk.?
    ADD N3 N5 = SUCC (SUCC (SUCC N5))

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  217. λ
    JS add = n => k => ?
    ADD := nk.?
    ADD N3 N5 = (SUCC ∘ SUCC ∘ SUCC) N5

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  218. λ
    JS add = n => k => ?
    ADD := nk.?
    ADD N3 N5 = N3 SUCC N5

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  219. λ
    JS add = n => k => n(succ)(k)
    ADD := nk.n SUCC k
    ADD N3 N5 = N3 SUCC N5

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  220. ADD N3 N5 = N3 SUCC N5
    = THRICE SUCC FIVEFOLD
    = SUCC (SUCC (SUCC FIVEFOLD)))
    = EIGHTFOLD

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  221. CHURCH ARITHMETIC
    Name -Calculus Use
    SUCC nf.B f (nf) = nfa.f(nfa) successor of n
    ADD nk.n SUCC k = nkf.B (n f) (k f) addition of n and k
    MULT nkf.n(kf) = B multiplication of n and k
    POW nk.kn = Th
    raise n to the power of k
    PRED n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0 predecessor of n
    PRED n.FST (n Φ (PAIR N0 N0)) predecessor of n (easier)
    SUB nk.k PRED n subtract k from n

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  222. λ
    JS mult = n => k => ?
    MULT := nk.?
    MULT N2 N3 = N6

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  223. λ
    JS mult = n => k => ?
    MULT := nk.?
    MULT N2 N3 f a = N6 f a

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  224. λ
    JS mult = n => k => ?
    MULT := nk.?
    MULT N2 N3 f a = (f ∘ f ∘ f ∘ f ∘ f ∘ f) a

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  225. λ
    JS mult = n => k => ?
    MULT := nk.?
    MULT N2 N3 f a = ((f ∘ f ∘ f) ∘ (f ∘ f ∘ f)) a

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  226. λ
    JS mult = n => k => ?
    MULT := nk.?
    MULT N2 N3 f a = ((N3 f) ∘ (N3 f)) a

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  227. λ
    JS mult = n => k => ?
    MULT := nk.?
    MULT N2 N3 f a = N2 (N3 f) a

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  228. λ
    JS mult = n => k => ?
    MULT := nk.?
    MULT N2 N3 f a = N2 (N3 f) a

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  229. λ
    JS mult = n => k => ?
    MULT := nk.?
    MULT N2 N3 f = N2 (N3 f)

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  230. λ
    JS mult = n => k => n(k(f))
    MULT := nkf.n(kf)
    MULT N2 N3 f = N2 (N3 f)

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  231. MULT N2 N3 f = N2 (N3 f)
    = TWICE (THRICE f)
    = (f ∘ f ∘ f) ∘ (f ∘ f ∘ f)
    = SIXFOLD f

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  232. CHURCH ARITHMETIC
    Name -Calculus Use
    SUCC nf.B f (nf) = nfa.f(nfa) successor of n
    ADD nk.n SUCC k = nkf.B (n f) (k f) addition of n and k
    MULT nkf.n(kf) multiplication of n and k
    POW nk.kn = Th
    raise n to the power of k
    PRED n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0 predecessor of n
    PRED n.FST (n Φ (PAIR N0 N0)) predecessor of n (easier)
    SUB nk.k PRED n subtract k from n

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  233. λ
    JS
    MULT := nkf.n(kf)
    MULT N2 N3 f = N2 (N3 f)
    mult = n => k => n(k(f))

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  234. λ
    JS
    MULT := nkf.n(kf)
    MULT N2 N3 f = (N2 ∘ N3) f
    mult = n => k => n(k(f))

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  235. λ
    JS
    MULT := nkf.n(kf)
    MULT N2 N3 f = (N2 ∘ N3) f
    mult = n => k => n(k(f))

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  236. λ
    JS
    MULT := nkf.n(kf)
    MULT N2 N3 = N2 ∘ N3
    mult = n => k => n(k(f))

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  237. λ
    JS
    MULT := nkf.n(kf)
    MULT N2 N3 = B N2 N3
    mult = n => k => n(k(f))

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  238. λ
    JS
    MULT := nkf.n(kf)
    MULT N2 N3 = B N2 N3
    mult = n => k => n(k(f))

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  239. λ
    JS
    MULT := nkf.n(kf)
    MULT = B
    mult = n => k => n(k(f))

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  240. MULT := B

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  241. Mult := n k f . n ( k f)
    = B = f g a . f ( g a)
    -EQUIVALENCE

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  242. CHURCH ARITHMETIC
    Name -Calculus Use
    SUCC nf.B f (nf) = nfa.f(nfa) successor of n
    ADD nk.n SUCC k = nkf.B (n f) (k f) addition of n and k
    MULT nkf.n(kf) = B multiplication of n and k
    POW nk.kn = Th
    raise n to the power of k
    PRED n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0 predecessor of n
    PRED n.FST (n Φ (PAIR N0 N0)) predecessor of n (easier)
    SUB nk.k PRED n subtract k from n

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  243. λ
    JS pow = n => k => ?
    POW := nk.?
    POW N2 N3 = N8

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  244. λ
    JS pow = n => k => ?
    POW := nk.?
    POW N2 N3 = N2 × N2 × N2

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  245. λ
    JS pow = n => k => ?
    POW := nk.?
    POW N2 N3 = N2 ∘ N2 ∘ N2

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  246. λ
    JS pow = n => k => ?
    POW := nk.?
    POW N2 N3 = N3 N2

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  247. λ
    JS pow = n => k => k(n)
    POW := nk.kn
    POW N2 N3 = N3 N2

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  248. POW N2 N3 = N3 N2
    = THRICE TWICE
    = TWICE ∘ TWICE ∘ TWICE
    = EIGHTFOLD

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  249. CHURCH ARITHMETIC
    Name -Calculus Use
    SUCC nf.B f (nf) = nfa.f(nfa) successor of n
    ADD nk.n SUCC k = nkf.B (n f) (k f) addition of n and k
    MULT nkf.n(kf) = B multiplication of n and k
    POW nk.kn raise n to the power of k
    PRED n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0 predecessor of n
    PRED n.FST (n Φ (PAIR N0 N0)) predecessor of n (easier)
    SUB nk.k PRED n subtract k from n

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  250. THRUSH
    af.fa

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  251. COMBINATORS
    Sym. Bird -Calculus Use Haskell
    I Idiot a.a identity id
    M Mockingbird f.ff self-application (cannot define)
    K Kestrel ab.a true, first, const const
    KI Kite ab.b = KI = CK false, second const id
    C Cardinal fab.fba reverse arguments flip
    B Bluebird fga.f(ga) 1°←1° composition (.)
    Th
    Thrush af.fa hold an argument
    V Vireo abf.fab = BCT hold a pair of args flip . flip id
    B1
    Blackbird fgab.f(gab) = BBB 1°←2° composition (.) . (.)

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  252. COMBINATORS
    Sym. Bird -Calculus Use Haskell
    I Idiot a.a identity id
    M Mockingbird f.ff self-application (cannot define)
    K Kestrel ab.a true, first, const const
    KI Kite ab.b = KI = CK false, second const id
    C Cardinal fab.fba reverse arguments flip
    B Bluebird fga.f(ga) 1°←1° composition (.)
    Th
    Thrush af.fa = CI hold an argument flip id
    V Vireo abf.fab = BCT hold a pair of args flip . flip id
    B1
    Blackbird fgab.f(gab) = BBB 1°←2° composition (.) . (.)

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  253. CHURCH ARITHMETIC
    Name -Calculus Use
    SUCC nf.B f (nf) = nfa.f(nfa) successor of n
    ADD nk.n SUCC k = nkf.B (n f) (k f) addition of n and k
    MULT nkf.n(kf) = B multiplication of n and k
    POW nk.kn raise n to the power of k
    PRED n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0 predecessor of n
    PRED n.FST (n Φ (PAIR N0 N0)) predecessor of n (easier)
    SUB nk.k PRED n subtract k from n

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  254. CHURCH ARITHMETIC
    Name -Calculus Use
    SUCC nf.B f (nf) = nfa.f(nfa) successor of n
    ADD nk.n SUCC k = nkf.B (n f) (k f) addition of n and k
    MULT nkf.n(kf) = B multiplication of n and k
    POW nk.kn = Th
    raise n to the power of k
    PRED n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0 predecessor of n
    PRED n.FST (n Φ (PAIR N0 N0)) predecessor of n (easier)
    SUB nk.k PRED n subtract k from n

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  255. ISZERO

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  256. IS0 N0 = T
    IS0 N1 = F
    IS0 N2 = F

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  257. λ
    JS is0 = n => ?
    IS0 := n.?

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  258. λ
    JS is0 = n => n(func)(arg)
    IS0 := n.n func arg

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  259. λ
    JS is0 = n => n(func)(arg)
    IS0 := n.n func arg
    N0

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  260. λ
    JS is0 = n => n(func)(T)
    IS0 := n.n func T
    N0

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  261. λ
    JS is0 = n => n(func)(T)
    IS0 := n.n func T
    N1
    F

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  262. λ
    JS is0 = n => n(func)(T)
    IS0 := n.n func T
    N2
    F

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  263. λ
    JS is0 = n => n(func)(T)
    IS0 := n.n func T
    N > 0
    F

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  264. λ
    JS is0 = n => n(K(F))(T)
    IS0 := n.n (KF) T

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  265. λ
    JS is0 = n => n(K(F))(T)
    IS0 := n.n (KF) T
    IS0 N3 = KF(KF(KF(T))) = F

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  266. λ
    JS is0 = n => n(K(F))(T)
    IS0 := n.n (KF) T
    IS0 N3 = KF(KF(KF(T))) = F

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  267. λ
    JS is0 = n => n(K(F))(T)
    IS0 := n.n (KF) T
    IS0 N3 = KF(KF(KF(T))) = F

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  268. CHURCH ARITHMETIC: BOOLEAN OPS
    Name -Calculus Use
    IS0 n.n (K F) T test if n = 0
    LEQ nk.IS0 (SUB n k) test if n <= k
    EQ nk.AND (LEQ n k) (LEQ k n) test if n = k
    GT nk.B1
    NOT LEQ test if n > k

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  269. + × ^

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  270. – 1 ?

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  271. PRED := n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0

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  272. N0: N0 (g.IS0 (g N1) ) (K N0) N0
    I
    (B SUCC g)
    N0
    N1: N1 (g.IS0 (g N1) ) (K N0) N0
    I
    (B SUCC g)
    N0
    N2: N2 (g.IS0 (g N1) ) (K N0) N0
    I
    (B SUCC g)
    N1
    N3: N3 (g.IS0 (g N1) ) (K N0) N0
    I
    (B SUCC g)
    N2
    (K N0)
    I
    (SUCC ∘ I)
    PRED := n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0

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  273. PAIR FST SND PHI

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  274. VIREO
    abf.fab

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  275. λ
    JS V = a => b => f => f(a)(b)
    V := abf.fab

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  276. λ
    JS V(I)(M)
    // f => f(I)(M)
    V I M
    = (f.f I M)
    V := abf.fab
    V = a => b => f => f(a)(b)

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  277. λ
    JS V(I)(M)(K)
    // (f => f(I)(M))(K) === ?
    V I M K
    = (f.f I M) K = ?
    V := abf.fab
    V = a => b => f => f(a)(b)

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  278. λ
    JS V(I)(M)(K)
    // (f => f(I)(M))(K) === I
    V I M K
    = (f.f I M) K = I
    V := abf.fab
    V = a => b => f => f(a)(b)

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  279. λ
    JS
    V I M KI
    = (f.f I M) KI = M
    V := abf.fab
    V = a => b => f => f(a)(b)
    V(I)(M)(KI)
    // (f => f(I)(M))(KI) === M

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  280. View Slide

  281. COMBINATORS
    Sym. Bird -Calculus Use Haskell
    I Idiot a.a identity id
    M Mockingbird f.ff self-application (cannot define)
    K Kestrel ab.a true, first, const const
    KI Kite ab.b = KI = CK false, second const id
    C Cardinal fab.fba reverse arguments flip
    B Bluebird fga.f(ga) 1°←1° composition (.)
    Th
    Thrush af.fa = CI hold an argument flip id
    V Vireo abf.fab = BCT hold a pair of args flip . flip id
    B1
    Blackbird fgab.f(gab) = BBB 1°←2° composition (.) . (.)

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  282. CHURCH PAIRS
    Sym. Name -Calculus Use
    PAIR abf.fab = V pair two arguments
    FST p.pK extract first of pair
    SND p.p(KI) extract second of pair
    Φ PHI p.PAIR (SND p) (SUCC (SND p) copy 2nd to 1st, inc 2nd
    SET1ST cp.PAIR c (SND p) set first, immutably
    SET2ND cp.PAIR (FST p) c set second, immutably

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  283. FST := p.pK
    SND := p.p(KI)

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  284. CHURCH PAIRS
    Sym. Name -Calculus Use
    PAIR abf.fab = V pair two arguments
    FST p.pK extract first of pair
    SND p.p(KI) extract second of pair
    Φ PHI p.PAIR (SND p) (SUCC (SND p) copy 2nd to 1st, inc 2nd
    SET1ST cp.PAIR c (SND p) set first, immutably
    SET2ND cp.PAIR (FST p) c set second, immutably

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  285. λ
    JS phi = p => pair (snd(p)) (succ(snd(p)))
    PHI := p.V (SND p) (SUCC (SND p))

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  286. Φ := p.PAIR (SND p) (SUCC (SND p))
    Φ (M, N7) = ?

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  287. Φ := p.PAIR (SND p) (SUCC (SND p))
    Φ (M, N7) = ( , )

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  288. Φ := p.PAIR (SND p) (SUCC (SND p))
    Φ (M, N7) = (N7, )

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  289. Φ := p.PAIR (SND p) (SUCC (SND p))
    Φ (M, N7) = (N7, N8)

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  290. Φ := p.PAIR (SND p) (SUCC (SND p))
    Φ (M, N7) = (N7, N8)
    Φ (N9, N2) = (N2, N3)

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  291. CHURCH PAIRS
    Sym. Name -Calculus Use
    PAIR abf.fab = V pair two arguments
    FST p.pK extract first of pair
    SND p.p(KI) extract second of pair
    Φ PHI p.PAIR (SND p) (SUCC (SND p) copy 2nd to 1st, inc 2nd
    SET1ST cp.PAIR c (SND p) set first, immutably
    SET2ND cp.PAIR (FST p) c set second, immutably

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  292. Φ := p.PAIR (SND p) (SUCC (SND p))
    Φ (M, N7) = (N7, N8)
    Φ (N9, N2) = (N2, N3)
    Φ (N0, N0) = (N0, N1)

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  293. Φ := p.PAIR (SND p) (SUCC (SND p))
    Φ (M, N7) = (N7, N8)
    Φ (N9, N2) = (N2, N3)
    Φ (N0, N0) = (N0, N1)
    Φ (N0, N1) = (N1, N2)

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  294. Φ := p.PAIR (SND p) (SUCC (SND p))
    Φ (M, N7) = (N7, N8)
    Φ (N9, N2) = (N2, N3)
    Φ (N0, N0) = (N0, N1)
    Φ (N0, N1) = (N1, N2)
    Φ (N1, N2) = (N2, N3)

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  295. Φ := p.PAIR (SND p) (SUCC (SND p))
    Φ (M, N7) = (N7, N8)
    Φ (N9, N2) = (N2, N3)
    Φ (N0, N0) = (N0, N1)
    Φ (N0, N1) = (N1, N2)
    Φ (N1, N2) = (N2, N3)
    N8 Φ (N0, N0) = ?

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  296. Φ := p.PAIR (SND p) (SUCC (SND p))
    Φ (M, N7) = (N7, N8)
    Φ (N9, N2) = (N2, N3)
    Φ (N0, N0) = (N0, N1)
    Φ (N0, N1) = (N1, N2)
    Φ (N1, N2) = (N2, N3)
    N8 Φ (N0, N0) = (N7, N8)

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  297. FST ( )
    Φ := p.PAIR (SND p) (SUCC (SND p))
    Φ (M, N7) = (N7, N8)
    Φ (N9, N2) = (N2, N3)
    Φ (N0, N0) = (N0, N1)
    Φ (N0, N1) = (N1, N2)
    Φ (N1, N2) = (N2, N3)
    N8 Φ (N0, N0) = FST (N7, N8) = N7

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  298. FST ( )
    Φ := p.PAIR (SND p) (SUCC (SND p))
    Φ (M, N7) = (N7, N8)
    Φ (N9, N2) = (N2, N3)
    Φ (N0, N0) = (N0, N1)
    Φ (N0, N1) = (N1, N2)
    Φ (N1, N2) = (N2, N3)
    N8 Φ (N0, N0) = FST (N7, N8) = N7

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  299. λ
    JS pred = n => fst(n(phi)(pair(n0)(n0)))
    PRED := n.FST (n Φ (PAIR N0 N0))

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  300. CHURCH ARITHMETIC
    Name -Calculus Use
    SUCC nf.B f (nf) = nfa.f(nfa) successor of n
    ADD nk.n SUCC k = nkf.B (n f) (k f) addition of n and k
    MULT nkf.n(kf) = B multiplication of n and k
    POW nk.kn = Th
    raise n to the power of k
    PRED n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0 predecessor of n
    PRED n.FST (n Φ (PAIR N0 N0)) predecessor of n (easier)
    SUB nk.k PRED n subtract k from n

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  301. SUB LEQ EQ GT

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  302. λ
    JS sub = n => k => k(pred)(n)
    SUB := nk.k PRED n

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  303. CHURCH ARITHMETIC
    Name -Calculus Use
    SUCC nf.B f (nf) = nfa.f(nfa) successor of n
    ADD nk.n SUCC k = nkf.B (n f) (k f) addition of n and k
    MULT nkf.n(kf) = B multiplication of n and k
    POW nk.kn = Th
    raise n to the power of k
    PRED n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0 predecessor of n
    PRED n.FST (n Φ (PAIR N0 N0)) predecessor of n (easier)
    SUB nk.k PRED n subtract k from n

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  304. λ
    JS leq = n => k => is0(sub(n)(k))
    LEQ := nk.IS0 (SUB n k)

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  305. λ
    JS eq = n => k => and(leq(n)(k))(leq(k)(n))
    EQ := nk.AND(LEQ n k)(LEQ k n)

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  306. CHURCH ARITHMETIC: BOOLEAN OPS
    Name -Calculus Use
    IS0 n.n (K F) T test if n = 0
    LEQ nk.IS0 (SUB n k) test if n <= k
    EQ nk.AND (LEQ n k) (LEQ k n) test if n = k
    GT nk.B1
    NOT LEQ test if n > k

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  307. λ
    JS eq = n => k => not(leq(n)(k))
    GT := nk.NOT (LEQ n k)

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  308. λ
    JS eq = n => k => not(leq(n)(k))
    GT := nk.NOT (LEQ n k)

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  309. λ
    JS eq = B(not)(leq) ???
    GT := B NOT LEQ ???

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  310. λ
    JS eq = B(not)(leq)
    GT := B NOT LEQ

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  311. λ
    JS eq = n => k => not(leq(n)(k))
    GT := nk.NOT (LEQ n k)

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  312. BLACKBIRD
    fgab.f(gab)

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  313. λ
    JS eq = B1(not)(leq)
    GT := B1
    NOT LEQ
    B1
    = fgab.f(gab)

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  314. CHURCH ARITHMETIC: BOOLEAN OPS
    Name -Calculus Use
    IS0 n.n (K F) T test if n = 0
    LEQ nk.IS0 (SUB n k) test if n <= k
    EQ nk.AND (LEQ n k) (LEQ k n) test if n = k
    GT nk.B1
    NOT LEQ test if n > k

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  315. COMBINATORS
    Sym. Bird -Calculus Use Haskell
    I Idiot a.a identity id
    M Mockingbird f.ff self-application (cannot define)
    K Kestrel ab.a true, first, const const
    KI Kite ab.b = KI = CK false, second const id
    C Cardinal fab.fba reverse arguments flip
    B Bluebird fga.f(ga) 1°←1° composition (.)
    Th
    Thrush af.fa = CI hold an argument flip id
    V Vireo abf.fab = BCT hold a pair of args flip . flip id
    B1
    Blackbird fgab.f(gab) 1°←2° composition

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  316. COMBINATORS
    Sym. Bird -Calculus Use Haskell
    I Idiot a.a identity id
    M Mockingbird f.ff self-application (cannot define)
    K Kestrel ab.a true, first, const const
    KI Kite ab.b = KI = CK false, second const id
    C Cardinal fab.fba reverse arguments flip
    B Bluebird fga.f(ga) 1°←1° composition (.)
    Th
    Thrush af.fa = CI hold an argument flip id
    V Vireo abf.fab = BCT hold a pair of args flip . flip id
    B1
    Blackbird fgab.f(gab) = BBB 1°←2° composition (.) . (.)

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  317. B C K I
    KI = K I = C K
    B1
    = B B B
    Th
    = C I
    V = B C Th
    = B C (C I)

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  318. QUESTION
    how many combinators

    are needed to form a basis?
    20? 10? 5?

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  319. STARLING · KESTREL
    S := abc.ac(bc)
    K := ab.a

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  320. THE
    SK
    COMBINATOR
    CALCULUS

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  321. I = S K K

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  322. I = S K K
    V = (S(K((S((S(K((

    S(KS))K)))S))(KK))))
    ((S(K(S((SK)K))))K)

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  323. seriously though,
    why?

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  324. View Slide

  325. View Slide

  326. View Slide

  327. View Slide


  328. ADDENDUM

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  329. COMBINATORS
    Sym. Bird -Calculus Use Haskell
    I Idiot a.a identity id
    M Mockingbird f.ff self-application (cannot define)
    K Kestrel ab.a true, first, const const
    KI Kite ab.b = KI = CK false, second const id
    C Cardinal fab.fba reverse arguments flip
    B Bluebird fga.f(ga) 1°←1° composition (.)
    Th
    Thrush af.fa = CI hold an argument flip id
    V Vireo abf.fab = BCT hold a pair of args flip . flip id
    B1
    Blackbird fgab.f(gab) = BBB 1°←2° composition (.) . (.)

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  330. CHURCH ENCODINGS: BOOLEANS
    Sym. Name -Calculus Use
    T TRUE ab.a = K = C(KI) encoding for true
    F FALSE ab.b = KI = CK encoding for false
    NOT p.pFT or C negation
    AND pq.pqF or pq.pqp conjunction
    OR pq.pTq or pq.ppq = M* disjunction
    BEQ pq.p q (NOT q) equality

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  331. CHURCH ENCODINGS: NUMERALS
    Sym. Name -Calculus Use
    N0 ZERO fa.a = F apply f no times to a
    N1 ONCE fa.f a = I* apply f once to a
    N2 TWICE fa.f (f a) apply 2-fold f to a
    N3 THRICE fa.f (f (f a)) apply 3-fold f to a
    N4 FOURFOLD fa.f (f (f (f a))) apply 4-fold f to a
    N5 FIVEFOLD fa.f (f (f (f (f a))))) apply 5-fold f to a

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  332. CHURCH ARITHMETIC
    Name -Calculus Use
    SUCC nf.B f (nf) = nfa.f(nfa) successor of n
    ADD nk.n SUCC k = nkf.B (n f) (k f) addition of n and k
    MULT nkf.n(kf) = B multiplication of n and k
    POW nk.kn = Th
    raise n to the power of k
    PRED n.n (g.IS0 (g N1) I (B SUCC g)) (K N0) N0 predecessor of n
    PRED n.FST (n Φ (PAIR N0 N0)) predecessor of n (easier)
    SUB nk.k PRED n subtract k from n

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  333. CHURCH ARITHMETIC: BOOLEAN OPS
    Name -Calculus Use
    IS0 n.n (K F) T test if n = 0
    LEQ nk.IS0 (SUB n k) test if n <= k
    EQ nk.AND (LEQ n k) (LEQ k n) test if n = k
    GT nk.B1
    NOT LEQ test if n > k

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  334. CHURCH PAIRS
    Sym. Name -Calculus Use
    PAIR abf.fab = V pair two arguments
    FST p.pK extract first of pair
    SND p.p(KI) extract second of pair
    Φ PHI p.PAIR (SND p) (SUCC (SND p) copy 2nd to 1st, inc 2nd
    SET1ST cp.PAIR c (SND p) set first, immutably
    SET2ND cp.PAIR (FST p) c set second, immutably

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  335. f.M(x.f(Mx))
    THE Y FIXED-POINT COMBINATOR

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  336. EVALUATION STRATEGIES
    (ab.b)((f.ff)f.ff) x
    = (b.b) x
    = x
    (ab.b)((f.ff)f.ff)x
    = (ab.b)((f.ff)f.ff)x
    = (ab.b)((f.ff)f.ff)x
    = (ab.b)((f.ff)f.ff)x
    = (ab.b)((f.ff)f.ff)x
    C A L L B Y N A M E
    (apply to args before reduction)
    C A L L B Y VA L U E
    (reduce args before application)
    (AKA normal order; lazy) (AKA applicative order; strict)

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  337. f.M(x.f(v.Mxv))
    THE Z FIXED-POINT COMBINATOR

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  338. ADDITIONAL RESOURCES
    Combinator Birds · Rathman · http://bit.ly/2iudab9
    To Mock a Mockingbird · Smullyan · http://amzn.to/2g9AlXl
    To Dissect a Mockingbird · Keenan · http://dkeenan.com/Lambda
    .:.

    A Tutorial Introduction to the Lambda Calculus · Rojas · http://bit.ly/1agRC97
    Lambda Calculus · Wikipedia · http://bit.ly/1TsPkGn
    The Lambda Calculus · Stanford · http://stanford.io/2vtg8hp
    .:.

    History of Lambda-calculus and Combinatory Logic
    Cardone, Hindley · http://bit.ly/2wCxv4k
    .:.

    An Introduction to Functional Programming

    through Lambda Calculus · Michaelson · http://amzn.to/2vtts56

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