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1 II
2
3 [A] D B A B A B A B
4
5 DVD
6 y = 2x + 5 x = 3 y = 11 x = 5 y = 15.
7 Google Web
8
9 (2 + 3)
10
11
12 Windows Media Player Media Player
13
14 (typed lambda calculus)
15 (computer science)
16
17
18
19 f(x) = (x + 3) 5 (x + 3) 5
20 (2 + 3) ! (2+3)!5
21 (x + 3) ! ( +3)!5
22 (x + 3) 5 x 3 5 +! (x+3)!5
23 (x + 3) 5
24 (x + 3) 5 x 3 5 +! (x+3)!5
25 x 3 5 +! (x+3)!5
26 λx.((x + 3) 5)
27 λx.((x + 3) 5) x 3 5 +! (x+3)!5
28 λx.((x + 3) 5) x (x + 3) 5
29 x 3 5 +! (x+3)!5
30 (x + 3) 5 x
31 (x + 3) 5 x x
32 (x + 3) 5 x (x + 3) 5 (x + 3) 5 x
33 x x : int x x int integer int
34 (x + 3) 5 x (x + 3) 5 : int x : int
35 λx.((x + 3) 5) x 3 5 +! (x+3)!5
36 λx.((x + 3) 5) x (x + 3) 5 λx.((x + 3) 5) : int int
37 λx.((x + 3) 5) : int int A B A B
38 x (x + 3) 5 x : int (x + 3) 5 : int λx.((x + 3) 5) : int int
39 [x : int]. (x + 3) 5 : int λx.((x + 3) 5) : int int λx.((x + 3) 5)
40 [x : A] D M : B λx.m : A B (abs) x A B M λx.m
41 [x : A] D M : B λx.m : A B (abs) λx.m A B A B x : A
42 [x : A] D M : B λx.m : A B (abs) (λ-abstraction)
43
44 λx.((x + 3) 5) : int int 2 (λx.((x + 3) 5)) 2 λx.((x + 3) 5) 2
45 2 (λx.((x + 3) 5)) 2 λx.((x + 3) 5) (λx.((x + 3) 5)) λx.((x + 3) 5)
46 2 (λx.((x + 3) 5)) M N M M M : A B 2 : int
47 (λx.((x + 3) 5)) λx.((x + 3) 5) λx.((x + 3) 5) : int int
48 M : A B N : A (app) MN : B M A B A B
49 M : A B N : A (app) MN : B
50
51 int (char) (Boolean) int int int (int int) (int int) (int int)
52 + + : int (int int) : int (int int)
53 [x : A] D M : B λx.m : A B M : A B N : A MN : B
54 λx.((x + 3) 5) + : i (i i) [x : i] x + : i i 3 : i : i (i i) (x + 3) : i (x + 3) : i i 5 : i ((x + 3) 5) : i λx.((x + 3) 5) : i i int i x 3 5 +! (x+3)!5
55
56 λx.((x + 3) 5) (typed lambda calculus)
57
58
59 (λx.((x + 3) 5)) 2
60 λx.((x + 3) 5) 3 5 +! ( +3)!5
61 (λx.((x + 3) 5)) 2 2 ( +3)!5 + 3! 5
62 (λx.((x + 3) 5)) 2 (2 + 3) ! (2+3)!5
63 x (λx.((x + 3) 5)) 2 (2 + 3) 5 = 25
64 (λx.((x + 3) 5)) 2 (2 + 3) 5 = 25 (λx.((x + 3) 5)) 3 (3 + 3) 5 = 30 (λx.((x + 3) 5)) 4 (4 + 3) 5 = 35.
65 β (λx.m)n M[x := N] (β-reduction)
66 [x : A] D M : B λx.m : A B M : A B N : A MN : B (λx.m)n M[x := N]
67 [x : A] D M : B λx.m : A B M : A B N : A MN : B (λx.m)n M[x := N]
68 .. λy.n 1 : A B. N 2 : A λx.m : B C (λy.n 1 )N 2 : B (λx.m)((λy.n 1 ) N 2 ) : C λy.n 1 N 2 λx.m
69 (λx.m)((λy.n 1 ) N 2 ) λy.n 1 N 2 λx.m 1 (λx.m)((λy.n 1 )N 2 ) (λx.m)(n 1 [y := N 2 ]) λx.m N 1 [y := N 2 ]
70 .. λx.m : B C N 1 [y := N 2 ] : B? (λx.m)(n 1 [y := N 2 ]) : C? N 1 [y := N 2 ] B λx.m
71 (λx.m)n : B M[x := N] : B
72 (λx.m)n M[x := N] (λx.m)n. D 1 λx.m : A B N : A (λx.m)n : B
73 (λx.m)n M[x := N] [x : A] D 0 M : B λx.m : A B (λx.m)n : B D 1 N : A
74 (λx.m)n M[x := N] [x : A] D 0 M : B λx.m : A B (λx.m)n : B D 1 N : A
75 (λx.m)n M[x := N] D 1 N : A D 0 [x := N] M[x := N] : B
76 [x : A] D 0 M : B λx.m : A B (λx.m)n : B D 1 N : A D 1 N : A D 0 [x := N] M[x := N] : B x N x N
77
78 x 3 5 +! (x+3)!5
79 [x : A] D M : B λx.m : A B M : A B N : A MN : B
80 + : i (i i) [x : i] x + : i i 3 : i : i (i i) (x + 3) : i (x + 3) : i i 5 : i ((x + 3) 5) : i λx.((x + 3) 5) : i i
81 (λx.m)n M[x := N] [x : A] D 0 M : B λx.m : A B (λx.m)n : B D 1 N : A D 1 N : A D 0 [x := N] M[x := N] : B
82 [x : A] D M : B λx.m : A B M : A B N : A MN : B (λx.m)n M[x := N]
83
84
85 Recall
86 [x : A] D M : B λx.m : A B [A] D B A B M : A B N : A MN : B A B A B
87
88 [x : A] D M : B λx.m : A B [A] D B A B λx.m
89 M : A B N : A MN : B A B A B M N
90 A x : A
91 [A (B C)] [A] [A B] [A] B C B C A C (A B) (A C) (A (B C)) ((A B) (A C))
92 [x : A (B C)] [y : A] [z : A B] [y : A] xy : B C zy : B (xy)(zy) : C λy.((xy)(zy)) : A C λzλy.((xy)(zy)) : (A B) (A C) λxλzλy.((xy)(zy)) : (A (B C)) ((A B) (A C))
93 = = =
94 (λx.m)n M[x := N] [x : A] D 0 M : B λx.m : A B (λx.m)n : B D 1 N : A D 1 N : A D 0 [x := N] M[x := N] : B
95 [x : A] D 0 M : B λx.m : A B (λx.m)n : B D 1 N : A D 1 N : A D 0 [x := N] M[x := N] : B
96 [A] D 0 B A B B D 1 A D 1 A D 0 B
97 [A] D 0 B A B B D 1 A
98 [A] D 0 B A B B D 1 A D 1 A A A
99 [A] D 0 B A B B D 1 A [A] D 0 B A B A B A B
100 [A] D 0 B A B B D 1 A A B A B
101 1. A 2. A B A B 3. A B A B B 1. A 2. A B B
102 [A] D 0 B A B B D 1 A D 1 A D 0 B A B
103 [A] D 0 B A B B D 1 A D 1 A D 0 B (reduction)
104
105 [A] D 0 B A B B D 1 A B
106 [A] D 0 B A B B D 1 A B A B A A B A B
107 [A] D 0 B A B B D 1 A A D 0 B B B A A B
108 [A] D 0 B A B B D 1 A D 1 A D 0 B B
109
110 [x : A] D M : B λx.m : A B [A] D B A B M : A B N : A MN : B A B A B
111 [x : A] D 0 M : B λx.m : A B (λx.m)n : B [A] D 0 B A B B D 1 N : A D 1 A D 1 N : A D 0 [x := N] M[x := N] : B D 1 A D 0 B
112 = = =
113
114 [2005]
115
116
117 [2005]
118
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