3. Çàâèñèìàÿ ôóíêöèÿ
(x : A) → B (x )
f : (x : Bool) -> (if x then String else Int)
f True = "42"
f False = 42
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4. Òèïû òèïîâ
òèï ÿâëÿåòñÿ âûðàæåíèåì
5 : Int
Int : Type
Int -> Int : Type
5 : (if False then String else Int)
(x : Bool) ->
(if x then String else Int) : Type
"What?" : if (isPrime 2147483647) then
String else Int
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7. Íàòóðàëüíûå ÷èñëà
data Nat : Type where
Z : Nat
S : Nat - Nat
(+) : Nat - Nat - Nat
Z
+ y = y
(S k) + y = S (k + y)
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8. Ñïèñêè ôèêñèðîâàííîé äëèííû
data List : Type - Type where
Nil : List a
(::) : a - List a - List a
data Vect : Type - Nat - Type where
Nil : Vect a Z
(::) : a - Vect a k - Vect a (S k)
app : Vect a n - Vect a m - Vect a (n + m)
app Nil ys = ys
app (x :: xs) ys = x :: app xs ys
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9. Ñïèñêè ôèêñèðîâàííîé äëèííû
data Vect : Type - Nat - Type where
Nil : Vect a Z
(::) : a - Vect a k - Vect a (S k)
Nil : Vect Int Z
(1 :: 2 :: 3 :: Nil) : Vect Int (S S S Z)
IntVect : Nat - Type
IntVect = Vect Int
(1 :: 2 :: Nil) : IntVect (S S Z)
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10. Ñïèñêè ôèêñèðîâàííîé äëèííû
data Vect : Type - Nat - Type where
Nil : Vect a Z
(::) : a - Vect a k - Vect a (S k)
head : Vect a (S k) - a
head (h :: t) = h
head (1 :: 2 :: 4 :: Nil)
head Nil
defaultHead : Vect a k - a - a
defaultHead Nil default = default
defaultHead v _ = head v
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11. Printf
sprintf Simple!!!
sprintf Hello %s!!! world
sprintf Answer is %d 42
sprintf Mistake %s %d 42 world
sprintf Forgot %d
sprintf Too much %d 1 2
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12. Printf
data Format = End |
FMInt Format |
FMString Format |
FMChar Char Format
parse
parse
parse
parse
parse
: List Char - Format
Nil = End
('%'::'d'::cs) = FMInt (parse cs)
('%'::'s'::cs) = FMString (parse cs)
(c::cs) = FMChar c (parse cs)
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14. sprintf : (f: String) -
ToType (parse (unpack f))
sprintf fmt = prt (parse (unpack f)) Nil
prt : (f: Format) - List Char - ToType f
prt End acc =
pack acc
prt (FMInt r) acc =
i: Int = prt r (acc ++ (unpack (show i)))
prt (FMString r) acc =
s: String = prt r (acc ++ (unpack s))
prt (FMChar c r) acc =
prt r (acc ++ [c])
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15. Çàâèñèìàÿ ïàðà
d : (x : Bool ** if x then String else Int)
d = (False ** 42)
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16. Êîíñòðóêòèâíàÿ ìàòåìàòèêà
data Or : a - b - Type where
Inl : a - Or a b
Inr : b - Or a b
data And : a - b - Type where
mkAnd : a - b - And
data True : Type
mkTrue : True
data _|_ : Type where
Not : Type - Type
Not a = a - _|_
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22. Ðàâåíòñâî
∀a ∈ T , ∀b ∈ T , ∀p (T ),
a = b → p(a) = p(b)
apply : (a : t) - (b : t) -
(p : t - t1) -
(a = b) - (p a) = (p b)
apply _ _ p (refl x) = refl (p x)
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23. Ïðèìåð
(+) : Nat - Nat - Nat
Z
+ y = y
(S k) + y = S (k + y)
∀a, b, c ∈ N, a + (b + c ) = (a + b) + c
assoc : (a : Nat) - (b : Nat) -
(a + (b + c)) = ((a + b)
assoc Z b c = refl (b + c)
assoc (S a) b c = apply (a + (b +
((a + b) + c) S (assoc a
(c : Nat) -
+ c)
c))
b c)
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24. Ïðîâåðêè âî âðåìÿ êîìïèëÿöèè
data so : Bool - Type where
oh : so True
test : Bool
test1 : so test
test1 = oh
test2 : so (2 * 2 == 4)
test2 = oh
testFail : so (2 * 2 == 5)
testFail = oh
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25. Ñòàòè÷åñêèå îãðàíè÷åíèÿ
data () : Nat - Nat - Type where
lessZ : Z (S k)
lessS : k1 k2 - (S k1) (S k2)
Array : Type - Nat - Type
get: (Array a sz) - (i: Nat ** i sz) - a
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26. lemma : i n - i (S n)
lemma lessZ = lessZ
lemma (lessS a) = lessS (lemma a)
range2 : (n : Nat) - (i : Nat) - i (S n) -
List (a : Nat ** a n)
range2 n Z lessZ = []
range2 n (S i) (lessS iln) =
(i ** iln) :: (range2 n i (lemma iln))
subList : (n : Nat) - (i : Nat) - (i S n) -
Array a n - List a
subList n i prf a = map (get a) (range2 n i prf)
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