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hierarchy.v
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(* monae: Monadic equational reasoning in Coq *)
(* Copyright (C) 2020 monae authors, license: LGPL-2.1-or-later *)
Ltac typeof X := type of X.
Require Import ssrmatching JMeq.
From mathcomp Require Import all_ssreflect ssralg ssrnum.
From mathcomp Require boolp.
From mathcomp Require Import mathcomp_extra reals.
From infotheo Require Import realType_ext.
Require Import preamble.
From HB Require Import structures.
(******************************************************************************)
(* A formalization of monadic effects over the category Set *)
(* *)
(* We consider the type Type of Coq as the category Set and define functors *)
(* and a hierarchy of monads on top of functors. These monads are used to *)
(* develop the basics of monadic equational reasoning. The file category.v *)
(* provides a more generic definition of functors and monads as well as a *)
(* bridge to this file. *)
(* *)
(* Module FunctorLaws == map laws of a functor *)
(* functor == type of functors *)
(* F # g == application of the functor F to the morphism g *)
(* F ~> G == natural transformation from functor F to functor G *)
(* f ~~> g == forall A, f A -> g A, notation used for the *)
(* components of a natural transformation *)
(* naturality F G f == the components f form a natural transformation *)
(* F ~> G *)
(* Module JoinLaws == join laws of a monad *)
(* isMonad == mixin of monad *)
(* >>= == notation for the standard bind operator *)
(* m >> f := m >>= (fun _ => f) *)
(* monad == type of monads *)
(* Ret == natural transformation idfun ~> M for a monad M *)
(* Join == natural transformation M \o M ~> M for a monad M *)
(* Module BindLaws == bind laws of a monad *)
(* *)
(* Failure and nondeterministic monads: *)
(* failMonad == failure monad *)
(* failR0Monad == fail is right zero of bind *)
(* altMonad == monad with nondeterministic choice *)
(* prePlusMonad == nondeterminism + failR0 + distributivity *)
(* plusMonad == preplusMonad + commutativity and idempotence *)
(* altCIMonad == altMonad + commutativity + idempotence *)
(* nondetMonad == failMonad + altMonad *)
(* nondetCIMonad == failMonad + altCIMonad *)
(* exceptMonad == failMonad + catch *)
(* *)
(* Control monads (wip): *)
(* contMonad, shiftresetMonad, jumpMonad *)
(* *)
(* State monads: *)
(* stateMonad S == state monad with a state of type S *)
(* stateRunMonad == state + RunStateT *)
(* let N be a monad and S be the type of states, then *)
(* when m is a computation in the monad *)
(* stateRunMonad S N, RunStateT m s runs m in a *)
(* state s and returns a computation in the monad N *)
(* exceptStateRunMonad == stateRun + except *)
(* reifyMonad == reify interface *)
(* stateReifyMonad == monadState + reify *)
(* failStateReifyMonad == stateReify + fail *)
(* nondetStateMonad == backtrackable state *)
(* arrayMonad == array monad *)
(* plusArrayMonad == plus monad + array monad *)
(* *)
(* ML_universe == a type with decidable equality to represent an *)
(* OCaml type together with its Coq representation *)
(* in the type of a Tarski universe *)
(* typedStoreMonad == A monad for OCaml computations *)
(* typedStoreRunMonad == typedStoreMonad + crun *)
(* *)
(* Trace monads: *)
(* traceMonad == trace monad *)
(* traceReifyMonad == trace + reify *)
(* stateTraceMonad == state + trace *)
(* stateTraceReifyMonad == stateTrace + reify *)
(* *)
(* Probability monads: *)
(* probMonad == probabilistic choice and bind left-distributes over *)
(* choice *)
(* probDrMonad == probMonad + bind right-distributes over choice *)
(* altProbMonad == combined (probabilistic and nondeterministic) choice *)
(* exceptProbMonad == exceptions + probabilistic choice *)
(* *)
(* Freshness monads: *)
(* freshMonad == monad with freshness *)
(* failFreshMonad == freshMonad + failure *)
(* *)
(* references: *)
(* - R. Affeldt, J. Garrigue, T. Saikawa, Environment-friendly monadic *)
(* equational reasoning for OCaml, The Coq Workshop 2023 *)
(* - R. Affeldt, D. Nowak, Extending Equational Monadic Reasoning with Monad *)
(* Transformers, TYPES 2020, https://arxiv.org/abs/2011.03463 *)
(* - R. Affeldt, D. Nowak, T. Saikawa, A Hierarchy of Monadic Effects for *)
(* Program Verification using Equational Reasoning, MPC 2019 *)
(* - J. Gibbons, R. Hinze, Just do it: simple monadic equational reasoning, *)
(* ICFP 2011 *)
(* - J. Gibbons, Unifying Theories of Programming with Monads, UTP 2012 *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Declare Scope do_notation.
Declare Scope mprog.
Declare Scope monae_scope.
Delimit Scope monae_scope with monae.
Declare Scope proba_monad_scope.
Reserved Notation "f (o) g" (at level 11).
Reserved Notation "m >> f" (at level 49).
Reserved Notation "'fmap' f" (at level 4).
Reserved Notation "x '[~]' y" (at level 50).
Notation "f ~~> g" := (forall A, f A -> g A)
(at level 51, only parsing) : monae_scope.
Local Open Scope monae_scope.
Notation UU1 := Type.
Notation UU0 := Type.
(* NB: not putting M in Set -> Set because of expressions like:
M (A * (size s).-1.-tuple A)%type *)
Module FunctorLaws.
Section def.
Variable (F : UU0 -> UU0) (f : forall A B : UU0, (A -> B) -> F A -> F B).
Definition id := forall A : UU0, f id = id :> (F A -> F A).
Definition comp := forall (A B C : UU0) (g : B -> C) (h : A -> B),
f (g \o h) = f g \o f h.
End def.
End FunctorLaws.
HB.mixin Record isFunctor (F : UU0 -> UU0) := {
actm : forall A B : UU0, (A -> B) -> F A -> F B ;
functor_id : FunctorLaws.id actm ;
functor_o : FunctorLaws.comp actm }.
#[short(type=functor)]
HB.structure Definition Functor := {F of isFunctor F}.
Notation "F # g" := (@actm F _ _ g) : monae_scope.
Notation "'fmap' f" := (_ # f) : mprog.
Section functorid.
Let id_actm (A B : UU0) (f : A -> B) : idfun A -> idfun B := f.
Let id_id : FunctorLaws.id id_actm. Proof. by []. Qed.
Let id_comp : FunctorLaws.comp id_actm. Proof. by []. Qed.
HB.instance Definition _ := isFunctor.Build idfun id_id id_comp.
End functorid.
Lemma FIdE (A B : UU0) (f : A -> B) : idfun # f = f. Proof. by []. Qed.
Section functor_composition.
Variables F G : functor.
Let comp_actm (A B : UU0) (h : A -> B) : (F \o G) A -> (F \o G) B :=
F # (G # h).
Let comp_id : FunctorLaws.id comp_actm.
Proof. by rewrite /FunctorLaws.id => A; rewrite /comp_actm 2!functor_id. Qed.
Let comp_comp : FunctorLaws.comp comp_actm.
Proof.
rewrite /FunctorLaws.comp => A B C g' h; rewrite /comp_actm.
by apply boolp.funext => m; rewrite [in RHS]compE 2!functor_o.
Qed.
HB.instance Definition _ := isFunctor.Build (F \o G) comp_id comp_comp.
End functor_composition.
Lemma FCompE (F G : functor) (A B : UU0) (k : A -> B) :
[the functor of F \o G] # k = F # (G # k).
Proof. by []. Qed.
(* monadic counterpart of function composition:
composes a pure function after a monadic function *)
Section fcomp.
Variable M : functor.
Definition fcomp (A B C : UU0) (f : A -> B) (g : C -> M A) :=
locked ((M # f) \o g).
Arguments fcomp : simpl never.
Local Notation "f (o) g" := (fcomp f g).
Lemma fcomp_def (A B C : UU0) (f : A -> B) (g : C -> M A) :
f (o) g = ((M # f)) \o g.
Proof. by rewrite /fcomp; unlock. Qed.
Lemma fcompE (A B C : UU0) (f : A -> B) (g : C -> M A) c :
(f (o) g) c = (M # f) (g c).
Proof. by rewrite fcomp_def. Qed.
Lemma fcomp_comp (A B C D : UU0) (f : A -> B) (g : C -> A) (m : D -> M C) :
(f \o g) (o) m = f (o) (g (o) m).
Proof. by rewrite 3!fcomp_def functor_o compA. Qed.
End fcomp.
Notation "f (o) g" := (fcomp f g) : mprog.
Arguments fcomp : simpl never.
Lemma functor_ext (F G : functor) :
forall (H : Functor.sort F = Functor.sort G),
@actm G =
eq_rect _ (fun m : UU0 -> UU0 => forall A B : UU0, (A -> B) -> m A -> m B)
(@actm F) _ H ->
G = F.
Proof.
move: F G => [F [[HF1 HF2 HF3]]] [G [[HG1 HG2 HG3]]] /= H.
subst F => /= H.
congr (Functor.Pack (Functor.Class _)).
have ? : HG1 = HF1.
rewrite /actm /= in H.
apply funext_dep => x.
apply funext_dep => y.
apply funext_dep => z.
by move/(congr1 (fun i => i x y z)) : H.
subst HG1.
congr (isFunctor.Axioms_ _); exact/proof_irr.
Defined.
Definition naturality (F G : functor) (f : F ~~> G) :=
forall (A B : UU0) (h : A -> B), (G # h) \o f A = f B \o (F # h).
Arguments naturality : clear implicits.
HB.mixin Record isNatural (F G : functor) (f : F ~~> G) := {
natural : naturality F G f }.
#[short(type=nattrans)]
HB.structure Definition Nattrans (F G : functor) := {f of isNatural F G f}.
Arguments natural {F G} s.
Notation "f ~> g" := (nattrans f g) : monae_scope.
Section natrans_lemmas.
Variables (F G : functor) (phi : F ~> G).
Lemma nattrans_ext (f g : F ~> G) : f = g <-> forall a, (f a = g a :> (_ -> _)).
Proof.
split => [ -> // |]; move: f g => [f Hf] [g Hg] /= fg.
have ? : f = g by exact: funext_dep.
subst g.
suff : Hf = Hg by move=> ->.
case: Hf => -[Hf].
case: Hg => -[Hg].
do 2 f_equal.
exact: proof_irr.
Qed.
End natrans_lemmas.
(*Require Import Logic.Eqdep.
Lemma natural_ext (F G G' : functor) (t : F ~> G) (t' : F ~> G') :
forall (H : G = G'),
forall (K : forall X (x : F X), Natural.cpnt t' x = eq_rect _ (fun m : functor => m X) (Natural.cpnt t x) _ H),
t' = eq_rect _ (fun m : functor => F ~> m) t _ H.
Proof.
move : t t' => [t t1] [t' t'1] /= H; subst G' => H /=.
have ? : t = t'.
apply funext_dep => A; apply funext_dep => x.
by rewrite H -[in RHS]eq_rect_eq.
subst t'.
congr Natural.Pack; exact/proof_irr.
Qed.
Lemma natural_ext2 (F F' : functor) (t : F \O F ~> F) (t' : F' \O F' ~> F') :
forall (K : F = F'),
forall L : (forall X (x : (F' \O F') X),
Natural.cpnt t' x = eq_rect _ (fun m : functor => m X)
(Natural.cpnt t (eq_rect _ (fun m : functor => (m \O m) X) x _ (esym K)))
_ K),
t' = eq_rect _ (fun m => m \O m ~> m) t _ K.
Proof.
move: t t' => [t t1] [t' t'1] /= H L; subst F.
rewrite -[in RHS]eq_rect_eq /=.
have ? : t = t'.
apply funext_dep => A; apply funext_dep => x.
by rewrite L -[in RHS]eq_rect_eq.
subst t'.
congr Natural.Pack; exact/proof_irr.
Qed.*)
Module JoinLaws.
Section join_laws.
Context {F : functor}.
Variables (ret : idfun ~~> F) (join : F \o F ~~> F).
Arguments ret {_}.
Arguments join {A}.
Definition left_unit := forall A, join \o ret = id :> (F A -> F A).
Definition right_unit := forall A, join \o F # ret = id :> (F A -> F A).
Definition associativity := forall A,
join \o F # join = join \o join :> (F (F (F A)) -> F A).
End join_laws.
End JoinLaws.
HB.mixin Record isMonad (F : UU0 -> UU0) of Functor F := {
ret : idfun ~> F ;
join : F \o F ~> F ;
bind : forall (A B : UU0), F A -> (A -> F B) -> F B ;
bindE : forall (A B : UU0) (f : A -> F B) (m : F A),
bind A B m f = join B ((F # f) m) ;
joinretM : JoinLaws.left_unit ret join ;
joinMret : JoinLaws.right_unit ret join ;
joinA : JoinLaws.associativity join }.
#[short(type=monad)]
HB.structure Definition Monad := {F of isMonad F &}.
(* we introduce Ret as a way to make the second arguments of ret implicit,
o.w. Coq won't let us *)
Notation Ret := (@ret _ _).
Notation Join := (@join _ _).
Arguments bind {s A B} : simpl never.
Notation "m >>= f" := (bind m f) : monae_scope.
Lemma eq_bind (M : monad) (A B : UU0) (m : M A) (f1 f2 : A -> M B) :
f1 =1 f2 -> m >>= f1 = m >>= f2.
Proof. by move=> f12; congr bind; apply boolp.funext. Qed.
Section monad_lemmas.
Variable M : monad.
Lemma fmapE (A B : UU0) (f : A -> B) (m : M A) :
(M # f) m = m >>= (ret B \o f).
Proof.
by rewrite bindE [in RHS]functor_o -[in RHS]compE compA joinMret.
Qed.
End monad_lemmas.
Module BindLaws.
Section bindlaws.
Variable F : UU0 -> UU0.
Variable b : forall (A B : UU0), F A -> (A -> F B) -> F B.
Local Notation "m >>= f" := (b m f).
Definition associative := forall A B C (m : F A) (f : A -> F B) (g : B -> F C),
(m >>= f) >>= g = m >>= (fun x => f x >>= g).
Definition right_distributive (add : forall B, F B -> F B -> F B) :=
forall A B (m : F A) (k1 k2 : A -> F B),
m >>= (fun x => add _ (k1 x) (k2 x)) = add _ (m >>= k1) (m >>= k2).
Definition left_distributive (add : forall B, F B -> F B -> F B) :=
forall A B (m1 m2 : F A) (k : A -> F B),
(add _ m1 m2) >>= k = add _ (m1 >>= k) (m2 >>= k).
Definition left_zero (f : forall A, F A) :=
forall (A B : UU0) (g : A -> F B), f A >>= g = f B.
Definition right_zero (f : forall A, F A) :=
forall (A B : UU0) (g : F B), g >>= (fun _ => f A) = f A.
Definition left_neutral (r : forall A : UU0, A -> F A) :=
forall (A B : UU0) (a : A) (f : A -> F B), r _ a >>= f = f a.
Definition right_neutral (r : forall A : UU0, A -> F A) :=
forall A (m : F A), m >>= r _ = m.
Definition left_id (r : forall A, F A) (op : forall B, F B -> F B -> F B) :=
forall A (m : F A), op _ (r _) m = m.
Definition right_id (r : forall A, F A) (op : forall B, F B -> F B -> F B) :=
forall A (m : F A), op _ m (r _) = m.
End bindlaws.
End BindLaws.
Definition join_of_bind (F : functor)
(b : forall (A B : UU0), F A -> (A -> F B) -> F B) : F \o F ~~> F :=
fun A : UU0 => (b _ A)^~ id.
Definition bind_of_join (F : functor) (j : F \o F ~~> F)
(A B : UU0) (m : F A) (f : A -> F B) : F B :=
j B ((F # f) m).
Section from_join_laws_to_bind_laws.
Variable (F : functor) (ret : idfun ~> F) (join : F \o F ~> F).
Hypothesis joinretM : JoinLaws.left_unit ret join.
Hypothesis joinMret : JoinLaws.right_unit ret join.
Hypothesis joinA : JoinLaws.associativity join.
Lemma bindretf_derived : BindLaws.left_neutral (bind_of_join join) ret.
Proof.
move=> A B a f; rewrite /bind_of_join -(compE (@join _)) -(compE _ (@ret _)).
by rewrite -compA (natural ret) compA joinretM compidf.
Qed.
Lemma bindmret_derived : BindLaws.right_neutral (bind_of_join join) ret.
Proof. by move=> A m; rewrite /bind_of_join -(compE (@join _)) joinMret. Qed.
Lemma bindA_derived : BindLaws.associative (bind_of_join join).
Proof.
move=> A B C m f g; rewrite /bind_of_join.
rewrite [LHS](_ : _ = ((@join _ \o (F # g \o @join _) \o F # f) m)) //.
rewrite (natural join) (compA (@join C)) -joinA -(compE (@join _)).
transitivity ((@join _ \o F # (@join _ \o (F # g \o f))) m) => //.
by rewrite -2!compA functor_o FCompE -[in LHS](@functor_o F).
Qed.
End from_join_laws_to_bind_laws.
HB.factory Record isMonad_ret_join (F : UU0 -> UU0) of isFunctor F := {
ret : idfun ~> F ;
join : F \o F ~> F ;
joinretM : JoinLaws.left_unit ret join ;
joinMret : JoinLaws.right_unit ret join ;
joinA : JoinLaws.associativity join }.
HB.builders Context M of isMonad_ret_join M.
Let F := [the functor of M].
Let bind (A B : UU0) (m : F A) (f : A -> F B) : F B :=
bind_of_join join m f.
Let bindE (A B : UU0) (f : A -> M B) (m : M A) :
bind m f = join B ((F # f) m).
Proof. by []. Qed.
HB.instance Definition _ := isMonad.Build M bindE joinretM joinMret joinA.
HB.end.
HB.factory Record isMonad_ret_bind (F : UU0 -> UU0) := {
ret : forall (A : UU0), A -> F A ;
bind : forall (A B : UU0), F A -> (A -> F B) -> F B ;
bindretf : BindLaws.left_neutral bind ret ;
bindmret : BindLaws.right_neutral bind ret ;
bindA : BindLaws.associative bind }.
HB.builders Context M of isMonad_ret_bind M.
Let actm (a b : UU0) (f : a -> b) m := bind m (@ret _ \o f).
Let actm_id : FunctorLaws.id actm.
Proof.
move=> a.
rewrite /actm; apply: boolp.funext => m /=.
by rewrite bindmret.
Qed.
Let actm_comp : FunctorLaws.comp actm.
Proof.
move=> a b c g h.
rewrite /actm; apply: boolp.funext => m /=.
rewrite bindA.
congr bind.
apply: boolp.funext => u /=.
by rewrite bindretf.
Qed.
HB.instance Definition _ := isFunctor.Build M actm_id actm_comp.
Let F := [the functor of M].
Local Notation FF := [the functor of F \o F].
Let ret_naturality : naturality idfun F ret.
Proof.
move=> a b h.
rewrite FIdE /hierarchy.actm /= /actm; apply: boolp.funext => m /=.
by rewrite bindretf.
Qed.
HB.instance Definition _ :=
isNatural.Build idfun F (ret : idfun ~~> F) ret_naturality.
Let join' : FF ~~> F := fun _ m => bind m idfun.
Let actm_bind (a b c : UU0) (f : a -> b) m (g : c -> F a) :
(actm f) (bind m g) = bind m (actm f \o g).
Proof. by rewrite /actm bindA. Qed.
Let join'_naturality : naturality FF F join'.
Proof.
move=> a b h.
rewrite /join' /=; apply: boolp.funext => mm /=.
rewrite /hierarchy.actm /= /isFunctor.actm /=.
rewrite actm_bind bindA /=.
congr bind.
apply: boolp.funext => m /=.
by rewrite bindretf /=.
Qed.
HB.instance Definition _ := isNatural.Build _ _ _ join'_naturality.
Let join := [the FF ~> F of join'].
Let bind_map (A B C : UU0) (f : A -> B) (m : M A) (g : B -> M C) :
bind ((F # f) m) g = bind m (g \o f).
Proof.
rewrite bindA; congr bind.
by apply: boolp.funext => ?; rewrite bindretf.
Qed.
Let bindE (a b : UU0) (f : a -> M b) (m : M a) :
bind m f = join b ((F # f) m).
Proof.
rewrite /join /= /hierarchy.actm /= /join' /=.
by rewrite bind_map.
Qed.
Let joinretM : JoinLaws.left_unit ret join.
Proof.
move=> a; apply: boolp.funext => m.
by rewrite /join /= /join' /= bindretf.
Qed.
Let joinMret : JoinLaws.right_unit ret join.
Proof.
move=> a; apply: boolp.funext => m.
rewrite /join /= /join'.
rewrite /hierarchy.actm /= /actm /=.
rewrite bindA /=.
rewrite [X in bind m X](_ : _ = fun x => ret x) ?bindmret //=; apply: boolp.funext => ?.
by rewrite bindretf.
Qed.
Let joinA : JoinLaws.associativity join.
Proof.
move => a; apply: boolp.funext => m.
rewrite /join /= /join'.
rewrite /hierarchy.actm /= /actm.
rewrite !bindA.
congr bind.
apply: boolp.funext => u /=.
by rewrite bindretf.
Qed.
HB.instance Definition _ := isMonad.Build M bindE joinretM joinMret joinA.
HB.end.
Section monad_lemmas.
Variable M : monad.
Lemma bindretf : BindLaws.left_neutral (@bind M) ret.
Proof.
move: (@bindretf_derived M ret join joinretM).
rewrite (_ : bind_of_join _ = @bind M) //.
apply funext_dep => A; apply funext_dep => B.
apply funext_dep => m; apply funext_dep => f.
by rewrite bindE.
Qed.
Lemma bindmret : BindLaws.right_neutral (@bind M) ret.
Proof.
move: (@bindmret_derived M ret join joinMret).
rewrite (_ : bind_of_join _ = @bind M) //.
apply funext_dep => A; apply funext_dep => B.
apply funext_dep => m; apply funext_dep => f.
by rewrite bindE.
Qed.
Lemma bindA : BindLaws.associative (@bind M).
Proof.
move: (@bindA_derived M join joinA).
rewrite (_ : bind_of_join _ = @bind M) //.
apply funext_dep => A; apply funext_dep => B.
apply funext_dep => m; apply funext_dep => f.
by rewrite bindE.
Qed.
End monad_lemmas.
Notation "'do' x <- m ; e" := (bind m (fun x => e)) (only parsing) : do_notation.
Notation "'do' x : T <- m ; e" :=
(bind m (fun x : T => e)) (only parsing) : do_notation.
Delimit Scope do_notation with Do.
Notation "m >> f" := (bind m (fun _ => f)) : monae_scope.
Fixpoint sequence (M : monad) A (s : seq (M A)) : M (seq A) :=
(if s isn't h :: t then Ret [::] else
do v <- h; do vs <- sequence t; Ret (v :: vs))%Do.
Lemma sequence_nil (M : monad) (A : UU0) :
sequence [::] = Ret [::] :> M (seq A).
Proof. by []. Qed.
Lemma sequence_cons (M : monad) A h (t : seq (M A)) :
(sequence (h :: t) = do x <- h ; do vs <- sequence t ; Ret (x :: vs))%Do.
Proof. by []. Qed.
Definition skip M := @ret M _ tt.
Arguments skip {M} : simpl never.
Ltac bind_ext :=
let congr_ext m := ltac:(congr (bind m); apply boolp.funext) in
match goal with
| |- @bind _ _ _ ?m ?f1 = @bind _ _ _ ?m ?f2 =>
congr_ext m
| |- @bind _ _ _ ?m1 ?f1 = @bind _ _ _ ?m2 ?f2 =>
first[ simpl m1; congr_ext m1 | simpl m2; congr_ext m2 ]
end.
Section bindskip.
Lemma bindmskip (M : monad) (m : M unit) : m >> skip = m.
Proof. rewrite -[RHS]bindmret; bind_ext; by case. Qed.
Lemma bindskipf (M : monad) A (m : M A) : skip >> m = m.
Proof. exact: bindretf. Qed.
End bindskip.
(* experimental *)
Tactic Notation "With" tactic(tac) "Open" ssrpatternarg(pat) :=
ssrpattern pat;
let f := fresh "f" in
intro f;
let g := fresh "g" in
let typ := typeof f in
let x := fresh "x" in
evar (g : typ);
rewrite (_ : f = g);
[rewrite {}/f {}/g|
apply boolp.funext => x; rewrite {}/g {}/f; tac]; last first.
Tactic Notation "Open" ssrpatternarg(pat) :=
With (idtac) Open pat.
Section fmap_and_join.
Variable M : monad.
Local Open Scope mprog.
Lemma bind_fmap (A B C : UU0) (f : A -> B) (m : M A) (g : B -> M C) :
fmap f m >>= g = m >>= (g \o f).
Proof. by rewrite fmapE bindA; under eq_bind do rewrite bindretf. Qed.
Lemma fmap_if (A B : UU0) (f : A -> B) b (m : M A) a :
fmap f (if b then m else Ret a) = if b then fmap f m else Ret (f a).
Proof. case: ifPn => Hb //; by rewrite fmapE bindretf. Qed.
Lemma fmap_bind (A B C : UU0) (f : A -> B) m (g : C -> M A) :
fmap f (m >>= g) = m >>= (f (o) g).
Proof.
rewrite fcomp_def fmapE bindA; bind_ext => c.
by rewrite compE -/(fmap _ _) fmapE.
Qed.
Lemma skip_fmap (A B : UU0) (f : A -> B) (mb : M B) ma :
mb >> (fmap f ma) = fmap f (mb >> ma).
Proof. by rewrite fmap_bind fcomp_def. Qed.
(*Lemma rev_map A B (f : A -> B) : rev \o map f = map f \o rev.
Proof.
apply functional_extensionality.
by elim=> // h t /= IH; rewrite !rev_cons IH map_rcons.
Qed.*)
Lemma joinE A (pp : M (M A)) : Join pp = pp >>= id.
Proof. rewrite bindE; congr Join; by rewrite functor_id. Qed.
Lemma join_fmap (A B : UU0) (f : A -> M B) (m : M A) : Join (fmap f m) = m >>= f.
Proof. by rewrite bindE. Qed.
End fmap_and_join.
Section kleisli.
Variable M : monad.
Implicit Types A B C D : UU0.
Definition kleisli A B C (m : B -> M C) (n : A -> M B) : A -> M C :=
Join \o (M # m) \o n.
Local Notation "m <=< n" := (kleisli m n).
Local Notation "m >=> n" := (kleisli n m).
Lemma kleisli_def A B C (g : B -> M C) (f : A -> M B) :
(f >=> g) = Join \o (M # g) \o f.
Proof. by []. Qed.
Lemma kleisliE A B C (g : B -> M C) (f : A -> M B) (a : A) :
(f >=> g) a = (f a) >>= g.
Proof. by rewrite /kleisli /= join_fmap. Qed.
Lemma bind_kleisli A B C m (f : A -> M B) (g : B -> M C) :
m >>= (f >=> g) = (m >>= f) >>= g.
Proof. by rewrite bindA; bind_ext => a; rewrite /kleisli !compE join_fmap. Qed.
Lemma ret_kleisli A B (k : A -> M B) : Ret >=> k = k.
Proof. by rewrite /kleisli -compA (natural ret) FIdE compA joinretM. Qed.
Local Open Scope mprog.
Lemma fcomp_kleisli A B C D (f : A -> B) (g : C -> M A) (h : D -> M C) :
f (o) (g <=< h) = (f (o) g) <=< h.
Proof.
rewrite /kleisli 2!fcomp_def 2!(compA (fmap f)) natural [in RHS]functor_o.
by rewrite -compA.
Qed.
Lemma kleisli_fcomp A B C (f : A -> M B) (g : B -> A) (h : C -> M B) :
((f \o g) <=< h) = f <=< (g (o) h).
Proof. by rewrite /kleisli fcomp_def functor_o 2!compA. Qed.
Local Close Scope mprog.
Lemma kleisliA A B C D (f : A -> M B) (g : B -> M C) (h : C -> M D) :
f >=> g >=> h = f >=> (g >=> h).
Proof.
apply: funext_dep => a; rewrite !kleisliE bindA.
by under [in RHS]eq_bind do rewrite kleisliE.
Qed.
End kleisli.
Notation "m <=< n" := (kleisli m n) : monae_scope.
Notation "m >=> n" := (kleisli n m) : monae_scope.
HB.mixin Record isMonadFail (M : UU0 -> UU0) of Monad M := {
fail : forall A : UU0, M A ;
(* exceptions are left-zeros of sequential composition *)
bindfailf : BindLaws.left_zero (@bind M) fail
(* fail A >>= f = fail B *) }.
#[short(type=failMonad)]
HB.structure Definition MonadFail := {M of isMonadFail M & }.
Arguments bindfailf [_].
Arguments fail {_} {_}.
Section guard_assert.
Variable M : failMonad.
Definition guard (b : bool) : M unit := if b then skip else fail.
Lemma guardPn (b : bool) : if_spec b skip fail (~~ b) b (guard b).
Proof. by rewrite /guard; case: ifPn => ?; constructor. Qed.
Lemma guardT : guard true = skip. Proof. by []. Qed.
Lemma guardF : guard false = fail. Proof. by []. Qed.
(* guard distributes over conjunction *)
Lemma guard_and a b : guard (a && b) = guard a >> guard b.
Proof.
by move: a b => -[|] b /=; [rewrite bindskipf | rewrite bindfailf].
Qed.
Definition assert {A : UU0} (p : pred A) (a : A) : M A := guard (p a) >> Ret a.
Lemma assertE {A : UU0} (p : pred A) (a : A) :
assert p a = guard (p a) >> Ret a.
Proof. by []. Qed.
Lemma assertT {A : UU0} (a : A) : assert xpredT a = Ret a.
Proof. by rewrite assertE guardT bindskipf. Qed.
Lemma assertF {A : UU0} (a : A) : assert xpred0 a = fail.
Proof. by rewrite assertE guardF bindfailf. Qed.
Lemma assertPn {A : UU0} (p : pred A) (a : A) :
if_spec (p a) (Ret a) fail (~~ (p a)) (p a) (assert p a).
Proof.
rewrite assertE; case: guardPn => pa;
by [rewrite bindskipf; constructor | rewrite bindfailf; constructor].
Qed.
Definition bassert {A : UU0} (p : pred A) (m : M A) : M A := m >>= assert p.
(* follows from guards commuting with anything *)
Lemma commutativity_of_assertions A q :
Join \o (M # (bassert q)) = bassert q \o Join :> (_ -> M A).
Proof.
by apply boolp.funext => x; rewrite !compE join_fmap /bassert joinE bindA.
Qed.
(* guards commute with anything *)
Lemma guardsC (HM : BindLaws.right_zero (@bind M) (@fail _)) b B (m : M B) :
guard b >> m = m >>= assert (fun=> b).
Proof.
case: guardPn => Hb.
- rewrite bindskipf.
under eq_bind do rewrite assertT.
by rewrite bindmret.
- under [RHS]eq_bind do rewrite assertF.
by rewrite bindfailf HM.
Qed.
End guard_assert.
Arguments assert {M} {A} : simpl never.
Arguments guard {M} : simpl never.
HB.mixin Record isMonadAlt (M : UU0 -> UU0) of Monad M := {
alt : forall T : UU0, M T -> M T -> M T ;
altA : forall T : UU0, associative (@alt T) ;
(* composition distributes leftwards over choice *)
alt_bindDl : BindLaws.left_distributive (@bind M) alt
(* in general, composition does not distribute rightwards over choice *)
(* NB: no bindDr to accommodate both angelic and demonic interpretations of
nondeterminism *) }.
#[short(type=altMonad)]
HB.structure Definition MonadAlt := {M of isMonadAlt M & }.
Notation "a [~] b" := (@alt _ _ a b). (* infix notation *)
HB.mixin Record isMonadAltCI (M : UU0 -> UU0) of MonadAlt M := {
altmm : forall A : UU0, idempotent (@alt M A) ;
altC : forall A : UU0, commutative (@alt M A) }.
#[short(type=altCIMonad)]
HB.structure Definition MonadAltCI := {M of isMonadAltCI M & }.
Arguments altC {_} {_}.
Arguments altmm {_} {_}.
Section altci_lemmas.
Variable (M : altCIMonad).
Lemma altCA A : @left_commutative (M A) (M A) (fun x y => x [~] y).
Proof. by move=> x y z; rewrite altA altC altA altC (altC x). Qed.
Lemma altAC A : @right_commutative (M A) (M A) (fun x y => x [~] y).
Proof. move=> x y z; by rewrite altC altA (altC x). Qed.
Lemma altACA A : @interchange (M A) (fun x y => x [~] y) (fun x y => x [~] y).
Proof. move=> x y z t; rewrite !altA; congr (_ [~] _); by rewrite altAC. Qed.
End altci_lemmas.
HB.mixin Record isMonadNondet (M : UU0 -> UU0) of MonadFail M & MonadAlt M := {
altfailm : @BindLaws.left_id M (@fail M) (@alt M) ;
altmfail : @BindLaws.right_id M (@fail M) (@alt M) }.
#[short(type=nondetMonad)]
HB.structure Definition MonadNondet := {M of isMonadNondet M & }.
#[short(type=nondetCIMonad)]
HB.structure Definition MonadCINondet := {M of MonadAltCI M & MonadNondet M }.
Section nondet_big.
Variables (M : nondetMonad) (A : UU0).
HB.instance Definition _ :=
Monoid.isLaw.Build _ _ _ (@altA M A) (@altfailm _ _) (@altmfail _ _).
Lemma test_bigop n : \big[(@alt _ _)/fail]_(i < n) (fail : M A) = fail.
Proof.
elim: n => [|n IH]; first by rewrite big_ord0.
by rewrite big_ord_recr /= IH altmfail.
Qed.
End nondet_big.
HB.mixin Record isMonadFailR0 (M : UU0 -> UU0) of MonadFail M :=
{ bindmfail : BindLaws.right_zero (@bind [the monad of M]) (@fail _) }.
#[short(type=failR0Monad)]
HB.structure Definition MonadFailR0 := {M of isMonadFailR0 M & }.
HB.mixin Record isMonadPrePlus (M : UU0 -> UU0)
of MonadNondet M & MonadFailR0 M :=
{ alt_bindDr : BindLaws.right_distributive (@bind M) alt }.
#[short(type=prePlusMonad)]
HB.structure Definition MonadPrePlus := {M of isMonadPrePlus M & }.
#[short(type=plusMonad)]
HB.structure Definition MonadPlus := {M of MonadCINondet M & MonadPrePlus M}.
HB.mixin Record isMonadExcept (M : UU0 -> UU0) of MonadFail M := {
catch : forall A, M A -> M A -> M A ;
(* monoid *)
catchmfail : forall A, right_id fail (@catch A) ;
catchfailm : forall A, left_id fail (@catch A) ;
catchA : forall A, associative (@catch A) ;
(* unexceptional bodies need no handler *)
catchret : forall A x, @left_zero (M A) (M A) (Ret x) (@catch A)
(* NB: left-zero of sequential composition inherited from failMonad *) }.
#[short(type=exceptMonad)]
HB.structure Definition MonadExcept := {M of isMonadExcept M & }.
Arguments catch {_} {_}.
HB.mixin Record isMonadContinuation (M : UU0 -> UU0) of Monad M := {
(* NB: interface is wip *)
callcc : forall A B : UU0, ((A -> M B) -> M A) -> M A;
callcc0 : forall (A B : UU0) (g : (A -> M B) -> M A) (k : B -> M B),
@callcc _ _ (fun f => g (fun x => f x >>= k)) = @callcc _ _ g
(* see Sect. 7.2 of [Schrijvers, 19] *);
callcc1 : forall (A B : UU0) (m : M B),
@callcc _ _ (fun _ : B -> M A => m) = m
(* see Sect. 3.3 of [Wadler, 94] *);
callcc2 : forall (A B C : UU0) (m : M A) x (k : A -> B -> M C),
@callcc _ _ (fun f : _ -> M _ => m >>= (fun a => f x >>= (fun b => k a b)))
= @callcc _ _ (fun f : _ -> M _ => m >> f x) ;
callcc3 : forall (A B : UU0) (m : M A) b,
@callcc _ _ (fun f : B -> M B => m >> f b) =
@callcc _ _ (fun _ : B -> M B => m >> Ret b) }.
#[short(type=contMonad)]
HB.structure Definition MonadContinuation := {M of isMonadContinuation M & }.
Arguments callcc {_} {_} {_}.
HB.mixin Record isMonadShiftReset (U : UU0) (M : UU0 -> UU0)
of MonadContinuation M := {
shift : forall A : UU0, ((A -> M U) -> M U) -> M A ;
reset : M U -> M U ;
shiftreset0 : forall (A : UU0) (m : M A), @shift _ (fun k => m >>= k) = m ;
(* see Sect. 3.3 of [Wadler, 94] *)
shiftreset1 : forall (A B : UU0) (h : (A -> M B) -> M A), callcc h =
@shift _ (fun k' => h (fun x => @shift _ (fun k'' => k' x)) >>= k') ;
(* see Sect. 3.3 of [Wadler, 94] *)
shiftreset2 : forall (A : UU0) (c : A) (c': U) (k : A -> U -> _),
(reset (do x <- Ret c; do y <- @shift _ (fun _ => Ret c'); k x y) =
Ret c >> Ret c')%Do ;
shiftreset3 : forall (c c' : U) (k : U -> U -> _),
(reset (do x <- Ret c; do y <- @shift _ (fun f => do v <- f c'; f v);
Ret (k x y)) =
reset (do x <- Ret c; do y <- @shift _ (fun f => f c');
Ret (k x (k x y))))%Do ;
shiftreset4 : forall (c : U) k,
(reset (do y <- @shift _ (@^~ c); Ret (k y)) = Ret (k c))%Do }.
#[short(type=shiftresetMonad)]
HB.structure Definition MonadShiftReset U := {M of isMonadShiftReset U M & }.
Arguments shift {_} {_} {_}.
(* NB: wip, no model *)
(* Sect. 7.2 of [Tom Schrijvers & al., Monad Transformers and Modular
Algebraic Effects: What Binds Them Together, Haskell 2019] *)
HB.mixin Record isMonadJump (ref : UU0 -> UU0) (M : UU0 -> UU0) of Monad M := {
jump : forall A B : UU0, ref A -> A -> M B;
sub : forall A B : UU0, (ref A -> M B) -> (A -> M B) -> M B;
jump0 : forall (A B : UU0) k x, @sub _ _ (fun r => @jump A B r x) k = k x ;
jump1 : forall (A B : UU0) p k, @sub A B (fun _ => p) k = p;
jump2 : forall (A B : UU0) p r', @sub _ _ p (@jump A B r') = p r';
jump3 : forall (A B : UU0) (p : ref A -> ref B -> M B) (k1 : A -> M B) k2,
@sub _ _ (fun r1 : ref A => @sub _ _ (fun r2 => p r1 r2) (k2 r1)) k1 =
@sub _ _ (fun r2 : ref B => @sub _ _ (fun r1 => p r1 r2) k1)
(fun x => @sub _ _ (k2^~ x) k1)
(*NB: differs from [Schrijvers et al. 19]*);
jump4 : forall (A B : UU0) r x k, (@jump A B r x) >>= k = @jump A B r x;
jump5 : forall (A B : UU0) p q k,
@sub A B p q >>= k = @sub A B (p >=> k) (q >=> k) }.
#[short(type=jumpMonad)]
HB.structure Definition MonadJump ref := {M of isMonadJump ref M & }.
Definition Jump ref (M : jumpMonad ref) := @jump ref M.
Arguments Jump {_} {_} {_} {_}.
Definition Sub ref (M : jumpMonad ref) := @sub ref M.
Arguments sub {_} {_} {_} {_}.
HB.mixin Record isMonadState (S : UU0) (M : UU0 -> UU0) of Monad M := {
get : M S ;
put : S -> M unit ;
putput : forall s s', put s >> put s' = put s' ;
putget : forall s, put s >> get = put s >> Ret s ;
getput : get >>= put = skip ;
getget : forall (A : UU0) (k : S -> S -> M A),
get >>= (fun s => get >>= k s) = get >>= fun s => k s s }.
#[short(type=stateMonad)]
HB.structure Definition MonadState (S : UU0) := { M of isMonadState S M & }.
(*NB: explicit join newly added*)
#[short(type=failStateMonad)]
HB.structure Definition MonadFailState (S : UU0) :=
{ M of isMonadFail M & isMonadState S M & isMonad M & isFunctor M }.
(*NB: explicit join newly added*)
#[short(type=failR0StateMonad)]
HB.structure Definition MonadFailR0State (S : UU0) :=
{ M of isMonadFailR0 M & isMonadState S M & isMonadFail M & isMonad M &
isFunctor M }.
#[short(type=nondetStateMonad)]
HB.structure Definition MonadNondetState (S : UU0) :=
{ M of MonadPrePlus M & MonadState S M }.
HB.mixin Record isMonadStateRun (S : UU0) (N : monad)
(M : UU0 -> UU0) of MonadState S M := {
runStateT : forall A : UU0, M A -> S -> N (A * S)%type ;
runStateTret : forall (A : UU0) (a : A) (s : S),
@runStateT _ (Ret a) s = Ret (a, s) ;
runStateTbind : forall (A B : UU0) (m : M A) (f : A -> M B) (s : S),
@runStateT _ (m >>= f) s =
@runStateT _ m s >>= fun x => @runStateT _ (f x.1) x.2 ;
runStateTget : forall s : S, @runStateT _ get s = Ret (s, s) ;
runStateTput : forall s' s : S, @runStateT _ (put s') s = Ret (tt, s') }.
#[short(type=stateRunMonad)]
HB.structure Definition MonadStateRun (S : UU0) (N : monad) :=
{M of isMonadStateRun S N M & }.
Arguments runStateT {_} {_} {_} {_}.
HB.mixin Record isMonadExceptStateRun
(S : UU0) (N : exceptMonad) (M : UU0 -> UU0)
of MonadStateRun S N M & MonadExcept M := Mixin {
runStateTfail : forall (A : UU0) (s : S),
runStateT (@fail [the exceptMonad of M] A) s = @fail N _ ;
runStateTcatch : forall (A : UU0) (s : S) (m1 m2 : M A),
runStateT (@catch [the exceptMonad of M] _ m1 m2) s =
@catch N _ (runStateT m1 s) (runStateT m2 s) }.
#[short(type=exceptStateRunMonad)]
HB.structure Definition MonadExceptStateRun (S : UU0) (N : exceptMonad) :=
{M of isMonadExceptStateRun S N M & }.
HB.mixin Record isMonadReify (S : UU0) (M : UU0 -> UU0) of Monad M := {