輪講 — 論理と計算のしくみ 5.3 型付きλ計算 (前半)

—
5.3 λ ( )
(id:tarao)
2012-01-26
1

, .
.
, 2007.
J. C. Mitchell.
Foundations for Programming Languages.
MIT Press, 1996.
B. C. Pierce.
Types and Programming Languages.
MIT Press, 2002.
2

(λx A→B .xx)(λx.xx)
◮ x
A → B
◮
A ◮
A → B (A × B )
◮ A → B ∈ A ? ?
⇒
?
3

[1/3]
◮
◮
◮
◮
◮
◮
4

[2/3]
◮ ,
◮ , ,
◮
◮ ,
◮ ,
◮
,
◮
◮ XSS,
5

[3/3]
◮ λ
( )
◮
◮ Gödel System T, Coq
◮ , ,
◮
◮ OCaml let rec
◮ Gradual Typing
6

λ
◮ (a), (b)
◮
◮
◮ 1 — (c)
◮
◮
◮ 2 —
(d)
◮
◮
7

(α, β, γ)
◮
(int )
◮
(A, B, C)
A ::= α | A → A
A ::= α | A → A | A × A | A + A
→
×, +
9

[1/2]
x A ⊢ M : A M A
⊢ x A : A
⊢ M : B
⊢ λx A .M : A → B
⊢ M : A ⊢ N : B
⊢ (M, N) : A × B
⊢ M : A → B ⊢ N : A
⊢ MN : B
⊢ L : A × B
π 1 (L) : A
⊢ L : A × B
π 2 (L) : B
10

[2/2] —
M (M is well-typed)
◮
⊢ M : A
λx A .x A x A
◮
11

—
◮
◮
→ β
◮ → β
Theorem (
(subject reduction))
⊢ M : A M → β N ⊢ N : A
Theorem ( R )
⊢ M : A M → β N , ⊢ N : B
BRA.
◮ R =
13

[1/2]
Lemma ( )
⊢ M : A x B ⊢ N : B
⊢ M[x := N] : A.
15

[2/2] — [1/3]
: M
◮ M = y
◮ y = x
1 y = x ⊢ y B : B
2 ⊢ M : A B = A
3 M[x := N] = N
4
B = A ⊢ N : A
◮ y ≠ x
1 y A 2 M[x := N] = y
3
⊢ y A : A
16

[2/2] — [2/3]
: M
◮ M = M 1 M 2
1
⊢ M 1 : C → A ⊢ M 2 : C
2 IH ⊢ M 1 [x := N] : C → A
3 IH ⊢ M 2 [x := N] : C
4 M 1 [x := N]M 2 [x := N] = (M 1 M 2 )[x := N]
5
⊢ (M 1 M 2 )[x := N] : A
17

[2/2] — [3/3]
: M
◮ M = λy A 1
.L
1 A = A 1 → A 2 ⊢ L : A 2
2 λz A 1
.L[y := z] = α λy A 1
.L
y ≠ x y N
3 λy A 1 .L[x := N] = (λyA 1
.L)[x := N]
4 IH ⊢ L[x := N] : A 2
5
⊢ (λy A 1 .L)[x := N] : A 1 → A 2
18

[1/2]
Theorem (
(subject reduction))
⊢ M : A M → β N ⊢ N : A
20

[2/2] — [1/2]
: M → β N
◮ Beta
1 M = (λx B .M 1 )M 2 → β M 1 [x := M 2 ] = N
2
⊢ M 1 : A ⊢ M 2 : B
3
⊢ M 1 [x := M 2 ] : A
◮ Lam
1 M = λx B .L → β λx B .L ′ = N L → β L ′
2 A = B → C ⊢ L : C
3 IH ⊢ L ′ : C
4
⊢ λx B .L ′ : B → C
21

[2/2] — [2/2]
: M → β N
◮ AppL
1 M = M 1 M 2 → β M 1M ′ 2 = N M 1 → β M 1
′
2
⊢ M 1 : B → A ⊢ M 2 : B
3 IH ⊢ M 1 ′ : B → A
4
⊢ M 1 ′ M 2 : A
◮ AppR
1 M = M 1 M 2 → β M 1 M 2 ′ = N M 2 → β M 2
′
2
⊢ M 1 : B → A ⊢ M 2 : B
3 IH ⊢ M 2 ′ : B
4
⊢ M 1 M 2 ′ : A
22

[1/4]
24

[2/4]
(λx.xx)(λx.xx)
◮
◮
◮
?
◮
?
◮
◮
25

[3/4] — [1/2]
β
M ∈ NF β ⇐⇒ M β
β
M ⇓ β N ⇐⇒
M = M 0 → β M 1 → β · · · → β M n = N ∈ NF β
26

[3/4] — [2/2]
Definition (
)
M , N
M ⇓ β N.
◮
1 ◮
Definition (
SN)
M , M
M → β M 1 → β M 2 → β · · ·
.
◮
27

[4/4]
◮
◮
⇐⇒
28

[1/2]
?
◮ M, N SN MN SN?
◮ M = λx.xx N = λx.xx SN
◮ Ω = MN
SN ◮ M, N
?
◮ OK
◮
⇒
29

[2/2]
◮ M = λx.xx SN MN SN
◮
◮ M “ ” : MN “ ”
30

1 “ ”
◮
◮
◮
SN
SN 2 “ ”
◮
=
2
31

1 “ ”
2 “ ”
◮ “ ”
◮ “ ” → β “ ”
◮ → β “ ”
“ ”
◮
3
“ ”
32

[1/6] —
R A
⊢ M : α M SN
R α (M)
⊢ M : A → B M SN
∀N.R A (N) ⇒ R B (MN)
R A→B (M)
◮ R A
SN
◮ R A
A
◮ cbv
M SN
M ⇓ cbv ∃ N
34

[2/6] — 5.16 [1/2]
Lemma (5.16)
0 ≤ i ≤ n.R Ai (M i )
R A (x A 1→···→A n →A M 1 · · · M n ).
◮ R A (x A )
◮ “ ”
35

[2/6] — 5.16 [2/2]
: A
1 R Ai (M i ) M i SN
2 x A 1→···→A n →A M 1 · · · M n SN
3 R Ai (M i ) ⊢ M i : A i
4
⊢ x A 1→···→A n →α M 1 · · ·M n : A
5 ◮ A = α
◮ R A (x A 1→···→A n →A M 1 · · ·M n )
◮ A = B → C
1 R B (N) N
2 IH R C (x A 1→···→A n →B→C M 1 · · · M n N)
3 R B→C (x A 1→···→A n →B→C M 1 · · ·M n )
36

[3/6] — 5.15 [1/3]
Lemma (5.15’)
M → β M ′ R A (M) R A (M ′ )
◮ ⇐=
( : M = (λxy.y)(λz.Ω) → β λy.y = M ′ )
◮ ⇐=
◮ ⇐=
◮ → cbv ⊢ M : A
OK [3]
37

[3/6] — 5.15 [2/3]
: A
◮ A = α
1 R α (M) M → β M ′
2 R α (M) ⊢ M : α M SN
3 M ′ SN( M )
◮ cbv
4
⊢ M ′ : α
5 R α (M ′ )
38

[3/6] — 5.15 [3/3]
: A
◮ A = B → C
1 R B→C (M) M → β M ′
2 N R B (N)
3 R B→C (M) R C (MN)
4 MN → β M ′ N IH R B (M ′ N)
5 R B (M ′ N) M ′ N SN
6 M ′ SN( M ′ N )
7
⊢ M ′ : B → C
8 R B→C (M ′ )
39

[4/6] — 5.17a [1/2]
Lemma (5.17a)
⊢ M 0 : A 1 → · · ·A n → α R B (L), R Ai (N i )
R α (M 0 [x B := L]N 1 · · · N n )
R α ((λx B .M 0 )LN 1 · · ·N n ).
◮
◮ (λx B .M 0 )LN 1 · · ·N n (λx B .M 0 )LN
40

[4/6] — 5.17a [2/2]
1 R B (L), R Ai (N i ) ⊢ L : B, ⊢ N i : A i
2
⊢ (λx B .M 0 )LN : α
3 R B (L), R Ai (N i ) L, N i SN
4 R α (M 0 [x B := L]N) M 0 [x B := L]N SN
5 M 0 , λx B .M 0 SN(
)
6 M 0 → ∗ β M 0, ′ L → ∗ β L′ , N i → ∗ β N i ′
M 0, ′ L ′ , N i ′
◮ M 0[x ′ := L ′ ]N ′ SN
◮ (λx B .M 0 )LN → ∗ β M 0[x ′ := L ′ ]N ′
7 (λx B M 0 )LN SN
8 R α ((λx B .M 0 )LN)
41

[5/6] — 5.17b [1/3]
Lemma (5.17b)
⊢ M 0 : A 1 → · · ·A n → A R B (L), R Ai (N i )
R A (M 0 [x B := L]N 1 · · ·N n )
R A ((λx B .M 0 )LN 1 · · ·N n )
◮
◮ 5.17a α A
42

[5/6] — 5.17b [2/3]
: A
◮ A = α
◮ 5.17a OK
43

[5/6] — 5.17b [3/3]
: A
◮ A = C 1 → C 2
1 M 1 R C1 (M 1 )
2 R C1 →C 2
(M 0 [x := L]N)
◮ ⊢ M 0 [x := L]N : C 1 → C 2
◮ R C2 (M 0 [x := L]NM 1 )
3 IH R C2 ((λx B .M 0 )LNM 1 )
◮ ⊢ (λx B .M 0 )LNM 1 : C 2
◮ (λx B .M 0 )LNM 1 SN
4
⊢ (λx B .M 0 )LN : C 1 → C 2
5 (λx B .M 0 )LN SN
44

[6/6] — 5.17 [1/4]
Lemma (5.17)
⊢ M : A R A1 (N 1 ), . . . , R An (N n )
R A (M[x A 1
1 := N 1 , . . ., x A n
n := N n]).
◮ x A 1
1 := N 1 , . . ., x A n
n
:= N n x := N
45

[6/6] — 5.17 [2/4]
M
◮ M = x i
1 M[x := N] = N i
2 R Ai (N i )
◮ M = y ≠ x i
1 M[x := N] = y A
2 5.16
46

[6/6] — 5.17 [3/4]
M
◮ M = M 1 M 2
1
⊢ M 1 : B → A ⊢ M 2 : B
2 IH R B→A (M 1 [x := N])
R B (M 2 [x := N])
3 R B→A
R A (M 1 [x := N]M 2 [x := N])
4 M 1 [x := N]M 2 [x := N] = (M 1 M 2 )[x := N]
5 R A ((M 1 M 2 )[x := N])
47

[6/6] — 5.17 [4/4]
M
◮ M = λx B .M 0
1
⊢ M 0 : C A = B → C
2
⊢ M 0 [x := N] : C
3 L R B (L)
4 IH R C (M 0 [x := N, x B := L])
5 5.17b R C ((λx B .M 0 [x := N])L)
6 M 0 [x := N, x B := L] SN
λx B .M 0 [x := N] SN
7 R B→C (λx B .M 0 [x := N])
48

Theorem (5.18)
λ
1 M ⊢ M : A
2 5.17 R A (M)
3 M SN
50