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  1. let f = fun(x) x in pair (f(3)) (f(true))
  2.  
  3. A = { 3:int, true:bool, pair: forall alf, beta. alf->beta->(alf x beta) }
  4.  
  5. AST:
  6.  
  7. Let:t12
  8. / | \
  9. / | \
  10. / | \
  11. f:* x:* apply:t11
  12. / \
  13. / \
  14. / \
  15. apply:t7 apply:t10
  16. / \ / \
  17. / \ / \
  18. / \ / \
  19. pair:<I> apply:t6 f:<I> true:bool
  20. / \
  21. / \
  22. f:<I> 3:int
  23.  
  24. *: GEN
  25. I: lookup(<ID>, A) and INST
  26.  
  27.  
  28. Derivation:
  29.  
  30. --------------------
  31. x:X |- x:X | {} (VAR)
  32. -------------------
  33. |- \x. x: X->X | {} (ABS) FTV(<empty env/context>) does not contain X
  34. -------------------
  35. |- (\x. x): forall X. X->X (GEN)
  36.  
  37. Hence, due to "Let f=fun(x) x in <body>", enter f: forall X. X->X into A.
  38.  
  39. A = { 3:int, true:bool, pair: forall alf, beta. alf->beta->(alf x beta), f: forall X. X->X }
  40.  
  41.  
  42. --------------- ---------------------
  43. A |- 3:int | {} A |- f:t41->t41 | {} (INST)
  44. ---------------------- ------------------- ------------------------------ --------------------------------------
  45. A |- true:bool | { } A |- f:t81->81 | {} A |- pair:a1->b1->(a1xb1) | {} A |- f 3:t6 | { t41->t41 = int -> t6 } (pair via INST)
  46. ------------------------------------------------------- -------------------------------------------------------------------------
  47. A |- f true :t10 | {t81->t81 = bool->t10} A |- pair f 3:t7 | { a1->b1->(a1xb1) = t6->t7 }
  48. ---------------------------------------------------------------------------------------------------------------------------------
  49. A |- pair (f 3) (f true):t11 | {t81->t81 = bool->t10, t41->t41 = int -> t6, a1->b1->(a1xb1) = t6->t7, t7 = t10->t11 }
  50. ---------------------------------------------------------------------------------------------------------------------------------
  51. A |- Let ... pair (f 4) (f true):t12 | {t81->t81 = bool->t10, t41->t41 = int -> t6, a1->b1->(a1xb1) = t6->t7, t7 = t10->t11, t12=t11 }
  52.  
  53.  
  54.  
  55. C = {t81->t81 = bool->t10, t41->t41 = int -> t6, a1->b1->(a1xb1) = t6->t7, t7=t10->t11, t12=t11}
  56. t = let f = \x.x in pair (f(3)) (f(true))
  57. S = t12
  58. T = SIGMA S
  59. X = { t81,t10,t41, t6, a1,b1, c1,t8, t11, t12}
  60.  
  61. Unification steps:
  62.  
  63. C = { t81 = bool, t81 = t10 , t41->t41 = int -> t6, a1->b1->(a1xb1) = t6->t7, t7=t10->t11, t12=t11 }
  64. SIGMA = {t81 |-> bool }
  65.  
  66. C = { bool = t10 , t41->t41 = int -> t6, a1->b1->(a1xb1) = t6->t7, t7=t10->t11, t12=t11 }
  67. SIGMA = { t10 |-> bool } o {t81 |-> bool }
  68.  
  69. C = { t41->t41 = int -> t6, a1->b1->(a1xb1) = t6->t7, t7=bool->t11, t12=t11 }
  70. C = { t41= int, t6 = int, a1->b1->(a1xb1) = t6->t7, t7=bool->t11, t12=t11 }
  71. SIGMA = { t41 |-> int} o { t6 |-> int} o { t10 |-> bool } o {t81 |-> bool }
  72. C = { a1->b1->(a1xb1) = int->t7, t7=bool->t11, t12=t11 }
  73.  
  74. C = { a1 = int, b1->(a1xb1) = t7, t7=bool->t11, t12=t11 }
  75. SIGMA = {a1 |-> int} o { t41 |-> int} o { t6 |-> int} o { t10 |-> bool } o {t81 |-> bool }
  76.  
  77. C = { b1->(intxb1) = t7, t7=bool->t11, t12=t11 }
  78. SIGMA = {t7 |-> b1->(intxb1) } o {a1 |-> int} o { t41 |-> int} o { t6 |-> int} o { t10 |-> bool } o {t81 |-> bool }
  79. C = { b1->(intxb1) = bool->t11, t12=t11 }
  80.  
  81. C = { b1 = bool, (intxb1) = t11, t12=t11 }
  82. SIGMA = {b1 |-> bool} o {t7 |-> b1->(intxb1) } o {a1 |-> int} o { t41 |-> int} o { t6 |-> int} o { t10 |-> bool } o {t81 |-> bool }
  83. C = { (intxbool) = t11, t12=t11 }
  84. SIGMA = { t11 |-> (intxbool) } o {b1 |-> bool} o {t7 |-> b1->(intxb1) } o {a1 |-> int} o { t41 |-> int} o { t6 |-> int} o { t10 |-> bool } o {t81 |-> bool }
  85. C = { (intxbool) = t12 }
  86. SIGMA = {t12 |-> (intxbool) } o { t11 |-> (intxbool) } o {b1 |-> bool} o {t7 |-> b1->(intxb1) } o {a1 |-> int} o { t41 |-> int} o { t6 |-> int} o { t10 |-> bool } o {t81 |-> bool }
  87.  
  88. T = SIGMA t12 = intxbool
  89.  
  90. Instead of using GEN and INST, you could also use LETPOLY (make copies), and ABSINF (assign new type to each copy).
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