Typeclass

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Introduction of type-classes in Coq

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Typeclass

  1. 1. Formal Methods Forum Type Class in Coq @tmiya April 20,2011@tmiya : Type Class in Coq, 1
  2. 2. PolymorphismPolymorphism “On Understanding Types, Data Abstraction, and Polymorphism” (Cardelli and Wegner, 1985) http://lucacardelli.name/Papers/Onunderstanding.a4.Pdf polymorphism = • universal: parametric : ML inclusion : • ad-hoc: overloading : Haskell Type class overloading coercion :@tmiya : Type Class in Coq, 2
  3. 3. Monad — from Hask : Haskell Hask : = = id= Hask m: return fmap ( ) Hask Hask = = return, fmap fmap f ma KS / mb O O return fmap return a /b f@tmiya : Type Class in Coq, 3
  4. 4. Monad (1)(2) bind (return’ a) f = f a µ ◦ ηT = 1T bind(·)return ma AI / ma O > T@ @@ }}} @@ @@ } @ @@ @@ }} bind }} @@ @@ } @@ @@ }} ηT @@ @@ }}} return @@ @@ }} @@ @@ }} a a T2 /T µ bind ma return’ = ma µ ◦ T η = 1T bind(·)f Tη ma / mb BJ / T2 O | T@ @@ | @@ @@ || @ @@ @@ || bind | @@ @@ || @@ @@ µ || f @@ return | @@ @@ @@ || @@ @@ ||| a b T@tmiya : Type Class in Coq, 4
  5. 5. Monad (3) bind (bind ma f) g µ ◦ µT = bind ma (fun a = bind (f a) g) = µ ◦ Tµ bind(·)f bind(·)g µT ma / / mc T3 / T2 O p7 mb 7 pp ppp ppppp ppppp µ ppp f ppp g return Tµ ppp ppp a b c T2 /T µ bind(·)(λa.bind(f (a))g ) /- ma o7 mb g gg3 mc bind(·)gg f ooooo gggg ggggg ooo ggggg ooo ggggg λa.bind(f (a))g a go ggg og c@tmiya : Type Class in Coq, 5
  6. 6. Monad M:Type-Type ( : M = option) A M A Haskell Coq Class Monad (M:Type - Type):Type := { return’ : forall {A}, A - M A; bind : forall {A B}, M A - (A - M B) - M B; left_unit : forall A B (a:A)(f:A - M B), bind (return’ a) f = f a; right_unit : forall A (ma:M A), bind ma return’ = ma; assoc : forall A B C (ma:M A)(f:A-M B)(g:B-M C), bind (bind ma f) g = bind ma (fun a = bind (f a) g) }.@tmiya : Type Class in Coq, 6
  7. 7. Haskell Haskell = do Notation m = f := (bind m f) (left associativity, at level 49). Notation ’do’ a - e ; c := (e = (fun a = c)) (at level 59, right associativity).@tmiya : Type Class in Coq, 7
  8. 8. Identity Identity M = id’ : A - A ( ) Qed. Defined. Definition id’(A:Type):Type := A. Instance Identity : Monad id’ := { return’ A a := a; bind A B ma f := f ma }. Proof. (* *) Defined.@tmiya : Type Class in Coq, 8
  9. 9. Maybe M = option : A - option A Maybe return’, bind Instance Maybe : Monad option := { return’ A a := Some a; bind A B ma f := match ma with | None = None | Some a = f a end }. Proof. (* *) Defined.@tmiya : Type Class in Coq, 9
  10. 10. Maybe Eval compute in (None = (fun x = Some (x + 1))). Eval compute in (Some 1 = (fun x = Some (x + 1))).@tmiya : Type Class in Coq, 10
  11. 11. List List flat_map (Hint: xs ++ app_ass ) Require Import List. Lemma flat_map_app : forall (A B:Type) (xs ys:list A)(f:A - list B), flat_map f (xs++ys) = flat_map f xs ++ flat_map f ys. Instance List : Monad list := { return’ A x := x :: nil; bind A B m f := flat_map f m }.@tmiya : Type Class in Coq, 11
  12. 12. MonadPlus Monad MonadPlus Class MonadPlus (M:Type - Type) : Type := { monad : Monad M ; (* MonadPlus *) }. OO Coq substructure (= ?) MonadPlus Monad monoid (monoid mzero mplus) bind mzero (mzero = 0, mplus = +, bind=× )@tmiya : Type Class in Coq, 12
  13. 13. MonadPlus Class MonadPlus (M:Type - Type) : Type := { monad : Monad M ; mzero : forall A, M A; mplus : forall A, M A - M A - M A; mzero_left : forall A f, bind (@mzero A) f = (@mzero A); mzero_right : forall A B (ma:M A), bind ma (fun x = (@mzero B)) = (@mzero B); monoid_left_unit : forall A (ma:M A), mplus _ (@mzero A) ma = ma; monoid_right_unit : forall A (ma:M A), mplus _ ma (@mzero A) = ma; monoid_assoc : forall A (ma mb mc:M A), mplus _ (mplus _ ma mb) mc = mplus _ ma (mplus _ mb mc) }.@tmiya : Type Class in Coq, 13
  14. 14. MaybePlus Maybe MaybePlus Instance MaybePlus : MonadPlus option := { monad := Maybe; mzero A := @None A; mplus A m1 m2 := match m1 with | None = m2 | Some x = m1 end }. Proof. (* *) Defined.@tmiya : Type Class in Coq, 14
  15. 15. ListPlus List ListPlus app_nil_end, app_ass@tmiya : Type Class in Coq, 15
  16. 16. A R A (CPS) Definition cont (R A:Type):Type := (A-R)-R. R False Require Import Logic.FunctionalExtensionality.@tmiya : Type Class in Coq, 16
  17. 17. Instance Cont R : Monad (cont R) := { return’ A a := fun k:A-R = k a; bind A B m f := fun k:B-R = m (fun a = f a k) }. Proof. (* left_unit *) intros A B a f. unfold cont in *. erewrite - eta_expansion. reflexivity. (* right_unit *) intros A ma. unfold cont in *. erewrite eta_expansion. extensionality x. erewrite - eta_expansion. auto. (* assoc *) intros A B C ma f g. reflexivity. Defined.@tmiya : Type Class in Coq, 17

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