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# Intro To Agda

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BAHUG meeting on February 22, 2010

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### Intro To Agda

1. 1. Intro to Agda by Larry Diehl (larrytheliquid)
2. 2. Why?
3. 3. induction centric data Nat : Set where   zero : Nat   suc : Nat -> Nat one = suc zero two = suc (suc zero)
4. 4. unicode centric data ℕ : Set where   zero : ℕ   suc : ℕ → ℕ suc₂ : ℕ → ℕ suc₂ n = suc (suc n)
5. 5. unicode in emacs r → bn ℕ :: ∷ member ∈ equiv ≡ and ∧ or ∨ neg ¬ ex ∃ all ∀
6. 6. built-ins data ℕ : Set where   zero : ℕ   suc : ℕ → ℕ {-# BUILTIN NATURAL ℕ #-} {-# BUILTIN ZERO zero #-} {-# BUILTIN SUC suc #-} nine = suc₂ 7
7. 7. standard library open import Data.Nat
8. 8. infix & pattern matching infixl 6 _+_ infixl 7 _*_ _+_ : ℕ → ℕ → ℕ zero + n = n suc m + n = suc (m + n) _*_ : ℕ → ℕ → ℕ zero * n = zero suc m * n = n + m * n
9. 9. emacs goal mode _+_ : ℕ → ℕ → ℕ  n+m=?   _+_ : ℕ → ℕ → ℕ n + m = {!!} _+_ : ℕ → ℕ → ℕ n + m = { }0 -- ?0 : ℕ
10. 10. coverage checking data Day : Set where   Sunday Monday Tuesday : Day   Wednesday Thursday : Day   Friday Saturday : Day   isWeekend : Day → Bool isWeekend Sunday = true isWeekend Saturday = true isWeekend _ = false
11. 11. termination checking (fail) isWeekend : Day → Bool isWeekend day = isWeekend day isWeekend : Day → Bool isWeekend Sunday = true isWeekend Saturday = isWeekend Sunday isWeekend _ = false
12. 12. termination checking (win) _+_ : ℕ → ℕ → ℕ zero + n = n suc m + n = suc (m + n) _*_ : ℕ → ℕ → ℕ zero * n = zero suc m * n = n + m * n
13. 13. data type polymorphism data List (A : Set) : Set where   []  : List A   _∷_ : (x : A) (xs : List A) → List A -- open import Data.List
14. 14. type-restriction append : List ℕ → List ℕ → List ℕ append [] ys = ys append (x ∷ xs) ys = x ∷ (append xs ys)
15. 15. parametric polymorphism append : (A : Set) → List A → List A → List A append A [] ys = ys append A (x ∷ xs) ys = x ∷ (append A xs ys)
16. 16. implicit arguments append : {A : Set} → List A → List A → List A append [] ys = ys append (x ∷ xs) ys = x ∷ (append xs ys) -- append = _++_
17. 17. indexed types data Vec (A : Set) : ℕ → Set where   []  : Vec A zero   _∷_ : {n : ℕ} (x : A) (xs : Vec A n) →            Vec A (suc n) bool-vec : Vec Bool 2 bool-vec = _∷_ {1} true    (_∷_ {0} false [])   bool-vec : Vec Bool 2 bool-vec = true ∷ false ∷ []
18. 18. indexed types append : {n m : ℕ} {A : Set} → Vec A n →   Vec A m → Vec A (n + m) append [] ys = ys append (x ∷ xs) ys = x ∷ (append xs ys)
19. 19. coverage checking head : {A : Set} → Vec A 9 → A head (x ∷ xs) = x   head : {n : ℕ} {A : Set} → Vec A (suc n) →  A head (x ∷ xs) = x   head : {n : ℕ} {A : Set} →   Vec A (1 + 1 * n) → A head (x ∷ xs) = x
20. 20. records record Car : Set where   field     make : Make     model : Model     year : ℕ electricCar : Car electricCar = record {     make = Tesla   ; model = Roadster   ; year = 2010  }
21. 21. records record Car : Set where   constructor _,_,_   field     make : Make     model : Model     year : ℕ electricCar : Car electricCar = Tesla , Roadster , 2010 carYear :  ℕ carYear = Car.year electricCar
22. 22. truth & absurdity record ⊤ : Set where data ⊥ : Set where T : Bool → Set T true = ⊤ T false = ⊥
23. 23. preconditions even : ℕ → Bool even zero = true even (suc zero) = false even (suc (suc n)) = even n evenSquare : (n : ℕ) → {_ : T (even  n)} → ℕ evenSquare n = n * n
24. 24. preconditions nonZero : ℕ → Bool nonZero zero = false nonZero _ = true postulate _div_ : ℕ → (n : ℕ) →    {_ : T (nonZero n)} → ℕ
25. 25. propositional equality infix 4 _≡_ data _≡_ {A : Set} (x : A) : A → Set where   refl : x ≡ x testMultiplication : 56 ≡ 8 * 7 testMultiplication = refl
26. 26. preconditions lookup : {A : Set}(n : ℕ)(xs : List A)   {_ : T (n < length xs)} → A lookup n []{()} lookup zero (x ∷ xs) = x lookup (suc n) (x ∷ xs) {p} = lookup n xs {p} testLookup : 2 ≡ lookup 1 (1 ∷ 2 ∷ []) testLookup = refl
27. 27. finite sets data Fin : ℕ → Set where   zero : {n : ℕ} → Fin (suc n)   suc  : {n : ℕ} (i : Fin n) → Fin (suc n) finTest : Fin 3 finTest = zero finTest₂ : Fin 3 finTest₂ = suc (suc zero)
28. 28. precondition with indexes lookup : {A : Set}{n : ℕ} → Fin n → Vec A n → A lookup zero (x ∷ xs) = x lookup (suc i) (x ∷ xs) = lookup i xs   testLookup : 2 ≡ lookup (suc zero) (1 ∷ 2 ∷ []) testLookup = refl testLookup₂ : 2 ≡ lookup (fromℕ 1) (1 ∷ 2 ∷ []) testLookup₂ = refl
29. 29. with doubleOrNothing :ℕ→ℕ doubleOrNothing n with even n doubleOrNothing n | true = n * 2 doubleOrNothing n | false = 0   doubleOrNothing :ℕ→ℕ doubleOrNothing n with even n ... | true = n * 2 ... | false = 0
30. 30. views parity : (n : ℕ) → Parity n parity zero = even zero parity (suc n) with parity n parity (suc .(k * 2)) | even k = odd k parity (suc .(1 + k * 2)) | odd k = even (suc k) half :ℕ→ℕ half n with parity n half .(k * 2) | even k = k half .(1 + k * 2) | odd k = k
31. 31. inverse data Image_∋_ {A B : Set}(f : A → B) :   B → Set where   im : {x : A} → Image f ∋ f x inv : {A B : Set}(f : A → B)(y : B) →   Image f ∋ y → A inv f .(f x) (im {x}) = x
32. 32. inverse Bool-to-ℕ : Bool → ℕ Bool-to-ℕ false = 0 Bool-to-ℕ true = 1 testImage : Image Bool-to-ℕ ∋ 0 testImage = im testImage₂ : Image Bool-to-ℕ ∋ 1 testImage₂ = im testInv : inv  Bool-to-ℕ 0 im ≡ false testInv = refl testInv₂ : inv  Bool-to-ℕ 1 im ≡ true testInv₂ = refl
33. 33. intuitionistic logic data _∧_ (A B : Set) : Set where   <_,_> : A → B → A ∧ B fst : {A B : Set} → A ∧ B → A fst < a , _ > = a snd : {A B : Set} → A ∧ B → B snd < _ , b > = b
34. 34. intuitionistic logic data _∨_ (A B : Set) : Set where   left : A → A ∨ B   right : B → A ∨ B case : {A B C : Set} → A ∨ B → (A → C) →   (B → C) → C case (left a) x _ = x a case (right b) _ y = y b
35. 35. intuitionistic logic record ⊤ : Set where   constructor tt data ⊥ : Set where exFalso : {A : Set} → ⊥ → A exFalso () ¬ : Set → Set ¬A=A→⊥
36. 36. intuitionistic logic _=>_ : (A B : Set) → Set A => B = A → B modusPonens : {A B : Set} → A => B → A → B modusPonens f a = f a _<=>_ : Set → Set → Set A <=> B = (A => B) ∧ (B => A)
37. 37. intuitionistic logic ∀′ : (A : Set)(B : A → Set) → Set ∀′ A B = (a : A) → B a data ∃ (A : Set)(B : A → Set) : Set where   exists : (a : A) → B a → ∃ A B witness : {A : Set}{B : A → Set} → ∃ A B → A witness (exists a _) = a
38. 38. intuitionistic logic data Man : Set where   person : Man isMan : Man → Set isMan m = ⊤ isMortal : Man → Set isMortal m = ⊤ socratesIsMortal : {Socrates : Man} →   (∀′ Man isMortal ∧ isMan Socrates) => isMortal Socrates socratesIsMortal {soc} < lemma , _ > = lemma soc
39. 39. The End Agda Wiki http://is.gd/8YgYM Dependently Typed Programming in Agda http://is.gd/8Ygyz Dependent Types at Work http://is.gd/8YggR lemmatheultimate.com