2018-02-14 17:33:01 +01:00
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From stdpp Require Import base.
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2018-02-14 12:55:20 +01:00
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Class Monad M {ret : MRet M} {bind : MBind M} :=
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{ ret_unit_1 : forall {A} (m : M A) , m ≫= mret = m
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; ret_unit_2 : forall {A B} (x : A) (f : A → M B),
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(mret x) ≫= f = f x
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; bind_assoc : forall {A B C} (m : M A) (f : A → M B) (g : B → M C),
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(m ≫= f) ≫= g = m ≫= (fun x => f x ≫= g)
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}.
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Global Instance Join_of_Monad M `{Monad M} : (MJoin M) :=
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fun A (m : M (M A)) => m ≫= id.
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Global Instance FMap_of_Monad M `{Monad M} : FMap M :=
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fun A B f ma => ma ≫= (fun x => mret (f x)).
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Class Monoid A (op: A → A → A) (e : A) :=
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{ op_assoc : Associative (=) op
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; e_left : LeftId (=) e op
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; e_right : RightId (=) e op
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}.
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Definition kleisli {A B C} `{mon : Monad M} (f : A → M B) (g : B → M C) (x : A) : M C := y ← f x; g y.
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Notation "f >=> g" := (kleisli f g) (at level 60, right associativity) : C_scope.
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Notation "( f >=>)" := (kleisli f) (only parsing) : C_scope.
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Notation "(>=> g )" := (fun f => kleisli f g) (only parsing) : C_scope.
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Notation "(>=>)" := (fun f g => kleisli f g) (only parsing) : C_scope.
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Notation "g <=< f" := (kleisli f g) (at level 60, right associativity) : C_scope.
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Notation "( g <=<)" := (fun f => kleisli f g) (only parsing) : C_scope.
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Notation "(<=< f )" := (kleisli f) (only parsing) : C_scope.
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