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Commit 45d4e2a6 authored by Robbert Krebbers's avatar Robbert Krebbers
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Turn the arguments of functors into COFEs. 

This allows one to make use of recursive ghost state obtained from the
recursive domain equation solver.
parent ccd42ca7
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Showing with 212 additions and 192 deletions
theories/algebra/agree.v
View file @ 45d4e2a6
......@@ -306,24 +306,24 @@ Proof.
Qed.
Program Definition agreeRF (F : cFunctor) : rFunctor := {|
rFunctor_car A B := agreeR (cFunctor_car F A B);
rFunctor_map A1 A2 B1 B2 fg := agreeC_map (cFunctor_map F fg)
rFunctor_car A _ B _ := agreeR (cFunctor_car F A B);
rFunctor_map A1 _ A2 _ B1 _ B2 _ fg := agreeC_map (cFunctor_map F fg)
|}.
Next Obligation.
intros ? A1 A2 B1 B2 n ???; simpl. by apply agreeC_map_ne, cFunctor_ne.
intros ? A1 ? A2 ? B1 ? B2 ? n ???; simpl. by apply agreeC_map_ne, cFunctor_ne.
Qed.
Next Obligation.
intros F A B x; simpl. rewrite -{2}(agree_map_id x).
intros F A ? B ? x; simpl. rewrite -{2}(agree_map_id x).
apply (agree_map_ext _)=>y. by rewrite cFunctor_id.
Qed.
Next Obligation.
intros F A1 A2 A3 B1 B2 B3 f g f' g' x; simpl. rewrite -agree_map_compose.
intros F A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x; simpl. rewrite -agree_map_compose.
apply (agree_map_ext _)=>y; apply cFunctor_compose.
Qed.
Instance agreeRF_contractive F :
cFunctorContractive F → rFunctorContractive (agreeRF F).
Proof.
intros ? A1 A2 B1 B2 n ???; simpl.
intros ? A1 ? A2 ? B1 ? B2 ? n ???; simpl.
by apply agreeC_map_ne, cFunctor_contractive.
Qed.
theories/algebra/auth.v
View file @ 45d4e2a6
......@@ -403,13 +403,13 @@ Proof. destruct x as [[[]|] ]; by rewrite // /auth_map /= agree_map_id. Qed.
Lemma auth_map_compose {A B C} (f : A → B) (g : B → C) (x : auth A) :
auth_map (g ∘ f) x = auth_map g (auth_map f x).
Proof. destruct x as [[[]|] ]; by rewrite // /auth_map /= agree_map_compose. Qed.
Lemma auth_map_ext {A B : ofeT} (f g : A → B) `{_ : NonExpansive f} x :
Lemma auth_map_ext {A B : ofeT} (f g : A → B) `{!NonExpansive f} x :
(∀ x, f x ≡ g x) → auth_map f x ≡ auth_map g x.
Proof.
constructor; simpl; auto.
apply option_fmap_equiv_ext=> a; by rewrite /prod_map /= agree_map_ext.
Qed.
Instance auth_map_ne {A B : ofeT} (f : A -> B) {Hf : NonExpansive f} :
Instance auth_map_ne {A B : ofeT} (f : A -> B) `{Hf : !NonExpansive f} :
NonExpansive (auth_map f).
Proof.
intros n [??] [??] [??]; split; simpl in *; [|by apply Hf].
......@@ -437,45 +437,45 @@ Proof. intros n f f' Hf [[[]|] ]; repeat constructor; try naive_solver;
apply agreeC_map_ne; auto. Qed.
Program Definition authRF (F : urFunctor) : rFunctor := {|
rFunctor_car A B := authR (urFunctor_car F A B);
rFunctor_map A1 A2 B1 B2 fg := authC_map (urFunctor_map F fg)
rFunctor_car A _ B _ := authR (urFunctor_car F A B);
rFunctor_map A1 _ A2 _ B1 _ B2 _ fg := authC_map (urFunctor_map F fg)
|}.
Next Obligation.
by intros F A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_ne.
by intros F A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply authC_map_ne, urFunctor_ne.
Qed.
Next Obligation.
intros F A B x. rewrite /= -{2}(auth_map_id x).
apply auth_map_ext=>y. apply _. apply urFunctor_id.
intros F A ? B ? x. rewrite /= -{2}(auth_map_id x).
apply (auth_map_ext _ _)=>y; apply urFunctor_id.
Qed.
Next Obligation.
intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -auth_map_compose.
apply auth_map_ext=>y. apply _. apply urFunctor_compose.
intros F A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x. rewrite /= -auth_map_compose.
apply (auth_map_ext _ _)=>y; apply urFunctor_compose.
Qed.
Instance authRF_contractive F :
urFunctorContractive F → rFunctorContractive (authRF F).
Proof.
by intros ? A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_contractive.
by intros ? A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply authC_map_ne, urFunctor_contractive.
Qed.
Program Definition authURF (F : urFunctor) : urFunctor := {|
urFunctor_car A B := authUR (urFunctor_car F A B);
urFunctor_map A1 A2 B1 B2 fg := authC_map (urFunctor_map F fg)
urFunctor_car A _ B _ := authUR (urFunctor_car F A B);
urFunctor_map A1 _ A2 _ B1 _ B2 _ fg := authC_map (urFunctor_map F fg)
|}.
Next Obligation.
by intros F A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_ne.
by intros F A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply authC_map_ne, urFunctor_ne.
Qed.
Next Obligation.
intros F A B x. rewrite /= -{2}(auth_map_id x).
apply auth_map_ext=>y. apply _. apply urFunctor_id.
intros F A ? B ? x. rewrite /= -{2}(auth_map_id x).
apply (auth_map_ext _ _)=>y; apply urFunctor_id.
Qed.
Next Obligation.
intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -auth_map_compose.
apply auth_map_ext=>y. apply _. apply urFunctor_compose.
intros F A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x. rewrite /= -auth_map_compose.
apply (auth_map_ext _ _)=>y; apply urFunctor_compose.
Qed.
Instance authURF_contractive F :
urFunctorContractive F → urFunctorContractive (authURF F).
Proof.
by intros ? A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_contractive.
by intros ? A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply authC_map_ne, urFunctor_contractive.
Qed.
theories/algebra/cmra.v
View file @ 45d4e2a6
......@@ -766,16 +766,18 @@ End cmra_morphism.
(** Functors *)
Record rFunctor := RFunctor {
rFunctor_car : ofeT → ofeT → cmraT;
rFunctor_map {A1 A2 B1 B2} :
rFunctor_car : ∀ A `{!Cofe A} B `{!Cofe B}, cmraT;
rFunctor_map `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} :
((A2 -n> A1) * (B1 -n> B2)) → rFunctor_car A1 B1 -n> rFunctor_car A2 B2;
rFunctor_ne A1 A2 B1 B2 :
NonExpansive (@rFunctor_map A1 A2 B1 B2);
rFunctor_id {A B} (x : rFunctor_car A B) : rFunctor_map (cid,cid) x ≡ x;
rFunctor_compose {A1 A2 A3 B1 B2 B3}
rFunctor_ne `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} :
NonExpansive (@rFunctor_map A1 _ A2 _ B1 _ B2 _);
rFunctor_id `{!Cofe A, !Cofe B} (x : rFunctor_car A B) :
rFunctor_map (cid,cid) x ≡ x;
rFunctor_compose `{!Cofe A1, !Cofe A2, !Cofe A3, !Cofe B1, !Cofe B2, !Cofe B3}
(f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x :
rFunctor_map (f◎g, g'◎f') x ≡ rFunctor_map (g,g') (rFunctor_map (f,f') x);
rFunctor_mor {A1 A2 B1 B2} (fg : (A2 -n> A1) * (B1 -n> B2)) :
rFunctor_mor `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2}
(fg : (A2 -n> A1) * (B1 -n> B2)) :
CmraMorphism (rFunctor_map fg)
}.
Existing Instances rFunctor_ne rFunctor_mor.
......@@ -785,13 +787,15 @@ Delimit Scope rFunctor_scope with RF.
Bind Scope rFunctor_scope with rFunctor.
Class rFunctorContractive (F : rFunctor) :=
rFunctor_contractive A1 A2 B1 B2 :> Contractive (@rFunctor_map F A1 A2 B1 B2).
rFunctor_contractive `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} :>
Contractive (@rFunctor_map F A1 _ A2 _ B1 _ B2 _).
Definition rFunctor_diag (F: rFunctor) (A: ofeT) : cmraT := rFunctor_car F A A.
Definition rFunctor_diag (F: rFunctor) (A: ofeT) `{!Cofe A} : cmraT :=
rFunctor_car F A A.
Coercion rFunctor_diag : rFunctor >-> Funclass.
Program Definition constRF (B : cmraT) : rFunctor :=
{| rFunctor_car A1 A2 := B; rFunctor_map A1 A2 B1 B2 f := cid |}.
{| rFunctor_car A1 _ A2 _ := B; rFunctor_map A1 _ A2 _ B1 _ B2 _ f := cid |}.
Solve Obligations with done.
Coercion constRF : cmraT >-> rFunctor.
......@@ -799,16 +803,18 @@ Instance constRF_contractive B : rFunctorContractive (constRF B).
Proof. rewrite /rFunctorContractive; apply _. Qed.
Record urFunctor := URFunctor {
urFunctor_car : ofeT → ofeT → ucmraT;
urFunctor_map {A1 A2 B1 B2} :
urFunctor_car : ∀ A `{!Cofe A} B `{!Cofe B}, ucmraT;
urFunctor_map `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} :
((A2 -n> A1) * (B1 -n> B2)) → urFunctor_car A1 B1 -n> urFunctor_car A2 B2;
urFunctor_ne A1 A2 B1 B2 :
NonExpansive (@urFunctor_map A1 A2 B1 B2);
urFunctor_id {A B} (x : urFunctor_car A B) : urFunctor_map (cid,cid) x ≡ x;
urFunctor_compose {A1 A2 A3 B1 B2 B3}
urFunctor_ne `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} :
NonExpansive (@urFunctor_map A1 _ A2 _ B1 _ B2 _);
urFunctor_id `{!Cofe A, !Cofe B} (x : urFunctor_car A B) :
urFunctor_map (cid,cid) x ≡ x;
urFunctor_compose `{!Cofe A1, !Cofe A2, !Cofe A3, !Cofe B1, !Cofe B2, !Cofe B3}
(f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x :
urFunctor_map (f◎g, g'◎f') x ≡ urFunctor_map (g,g') (urFunctor_map (f,f') x);
urFunctor_mor {A1 A2 B1 B2} (fg : (A2 -n> A1) * (B1 -n> B2)) :
urFunctor_mor `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2}
(fg : (A2 -n> A1) * (B1 -n> B2)) :
CmraMorphism (urFunctor_map fg)
}.
Existing Instances urFunctor_ne urFunctor_mor.
......@@ -818,13 +824,15 @@ Delimit Scope urFunctor_scope with URF.
Bind Scope urFunctor_scope with urFunctor.
Class urFunctorContractive (F : urFunctor) :=
urFunctor_contractive A1 A2 B1 B2 :> Contractive (@urFunctor_map F A1 A2 B1 B2).
urFunctor_contractive `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} :>
Contractive (@urFunctor_map F A1 _ A2 _ B1 _ B2 _).
Definition urFunctor_diag (F: urFunctor) (A: ofeT) : ucmraT := urFunctor_car F A A.
Definition urFunctor_diag (F: urFunctor) (A: ofeT) `{!Cofe A} : ucmraT :=
urFunctor_car F A A.
Coercion urFunctor_diag : urFunctor >-> Funclass.
Program Definition constURF (B : ucmraT) : urFunctor :=
{| urFunctor_car A1 A2 := B; urFunctor_map A1 A2 B1 B2 f := cid |}.
{| urFunctor_car A1 _ A2 _ := B; urFunctor_map A1 _ A2 _ B1 _ B2 _ f := cid |}.
Solve Obligations with done.
Coercion constURF : ucmraT >-> urFunctor.
......@@ -1189,16 +1197,16 @@ Proof.
Qed.
Program Definition prodRF (F1 F2 : rFunctor) : rFunctor := {|
rFunctor_car A B := prodR (rFunctor_car F1 A B) (rFunctor_car F2 A B);
rFunctor_map A1 A2 B1 B2 fg :=
rFunctor_car A _ B _ := prodR (rFunctor_car F1 A B) (rFunctor_car F2 A B);
rFunctor_map A1 _ A2 _ B1 _ B2 _ fg :=
prodC_map (rFunctor_map F1 fg) (rFunctor_map F2 fg)
|}.
Next Obligation.
intros F1 F2 A1 A2 B1 B2 n ???. by apply prodC_map_ne; apply rFunctor_ne.
intros F1 F2 A1 ? A2 ? B1 ? B2 ? n ???. by apply prodC_map_ne; apply rFunctor_ne.
Qed.
Next Obligation. by intros F1 F2 A B [??]; rewrite /= !rFunctor_id. Qed.
Next Obligation. by intros F1 F2 A ? B ? [??]; rewrite /= !rFunctor_id. Qed.
Next Obligation.
intros F1 F2 A1 A2 A3 B1 B2 B3 f g f' g' [??]; simpl.
intros F1 F2 A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' [??]; simpl.
by rewrite !rFunctor_compose.
Qed.
Notation "F1 * F2" := (prodRF F1%RF F2%RF) : rFunctor_scope.
......@@ -1207,21 +1215,21 @@ Instance prodRF_contractive F1 F2 :
rFunctorContractive F1 → rFunctorContractive F2 →
rFunctorContractive (prodRF F1 F2).
Proof.
intros ?? A1 A2 B1 B2 n ???;
intros ?? A1 ? A2 ? B1 ? B2 ? n ???;
by apply prodC_map_ne; apply rFunctor_contractive.
Qed.
Program Definition prodURF (F1 F2 : urFunctor) : urFunctor := {|
urFunctor_car A B := prodUR (urFunctor_car F1 A B) (urFunctor_car F2 A B);
urFunctor_map A1 A2 B1 B2 fg :=
urFunctor_car A _ B _ := prodUR (urFunctor_car F1 A B) (urFunctor_car F2 A B);
urFunctor_map A1 _ A2 _ B1 _ B2 _ fg :=
prodC_map (urFunctor_map F1 fg) (urFunctor_map F2 fg)
|}.
Next Obligation.
intros F1 F2 A1 A2 B1 B2 n ???. by apply prodC_map_ne; apply urFunctor_ne.
intros F1 F2 A1 ? A2 ? B1 ? B2 ? n ???. by apply prodC_map_ne; apply urFunctor_ne.
Qed.
Next Obligation. by intros F1 F2 A B [??]; rewrite /= !urFunctor_id. Qed.
Next Obligation. by intros F1 F2 A ? B ? [??]; rewrite /= !urFunctor_id. Qed.
Next Obligation.
intros F1 F2 A1 A2 A3 B1 B2 B3 f g f' g' [??]; simpl.
intros F1 F2 A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' [??]; simpl.
by rewrite !urFunctor_compose.
Qed.
Notation "F1 * F2" := (prodURF F1%URF F2%URF) : urFunctor_scope.
......@@ -1230,7 +1238,7 @@ Instance prodURF_contractive F1 F2 :
urFunctorContractive F1 → urFunctorContractive F2 →
urFunctorContractive (prodURF F1 F2).
Proof.
intros ?? A1 A2 B1 B2 n ???;
intros ?? A1 ? A2 ? B1 ? B2 ? n ???;
by apply prodC_map_ne; apply urFunctor_contractive.
Qed.
......@@ -1450,47 +1458,47 @@ Proof.
Qed.
Program Definition optionRF (F : rFunctor) : rFunctor := {|
rFunctor_car A B := optionR (rFunctor_car F A B);
rFunctor_map A1 A2 B1 B2 fg := optionC_map (rFunctor_map F fg)
rFunctor_car A _ B _ := optionR (rFunctor_car F A B);
rFunctor_map A1 _ A2 _ B1 _ B2 _ fg := optionC_map (rFunctor_map F fg)
|}.
Next Obligation.
by intros F A1 A2 B1 B2 n f g Hfg; apply optionC_map_ne, rFunctor_ne.
by intros F A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply optionC_map_ne, rFunctor_ne.
Qed.
Next Obligation.
intros F A B x. rewrite /= -{2}(option_fmap_id x).
intros F A ? B ? x. rewrite /= -{2}(option_fmap_id x).
apply option_fmap_equiv_ext=>y; apply rFunctor_id.
Qed.
Next Obligation.
intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -option_fmap_compose.
intros F A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x. rewrite /= -option_fmap_compose.
apply option_fmap_equiv_ext=>y; apply rFunctor_compose.
Qed.
Instance optionRF_contractive F :
rFunctorContractive F → rFunctorContractive (optionRF F).
Proof.
by intros ? A1 A2 B1 B2 n f g Hfg; apply optionC_map_ne, rFunctor_contractive.
by intros ? A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply optionC_map_ne, rFunctor_contractive.
Qed.
Program Definition optionURF (F : rFunctor) : urFunctor := {|
urFunctor_car A B := optionUR (rFunctor_car F A B);
urFunctor_map A1 A2 B1 B2 fg := optionC_map (rFunctor_map F fg)
urFunctor_car A _ B _ := optionUR (rFunctor_car F A B);
urFunctor_map A1 _ A2 _ B1 _ B2 _ fg := optionC_map (rFunctor_map F fg)
|}.
Next Obligation.
by intros F A1 A2 B1 B2 n f g Hfg; apply optionC_map_ne, rFunctor_ne.
by intros F A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply optionC_map_ne, rFunctor_ne.
Qed.
Next Obligation.
intros F A B x. rewrite /= -{2}(option_fmap_id x).
intros F A ? B ? x. rewrite /= -{2}(option_fmap_id x).
apply option_fmap_equiv_ext=>y; apply rFunctor_id.
Qed.
Next Obligation.
intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -option_fmap_compose.
intros F A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x. rewrite /= -option_fmap_compose.
apply option_fmap_equiv_ext=>y; apply rFunctor_compose.
Qed.
Instance optionURF_contractive F :
rFunctorContractive F → urFunctorContractive (optionURF F).
Proof.
by intros ? A1 A2 B1 B2 n f g Hfg; apply optionC_map_ne, rFunctor_contractive.
by intros ? A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply optionC_map_ne, rFunctor_contractive.
Qed.
(* Dependently-typed functions over a discrete domain *)
......@@ -1573,23 +1581,23 @@ Proof.
Qed.
Program Definition ofe_funURF {C} (F : C → urFunctor) : urFunctor := {|
urFunctor_car A B := ofe_funUR (λ c, urFunctor_car (F c) A B);
urFunctor_map A1 A2 B1 B2 fg := ofe_funC_map (λ c, urFunctor_map (F c) fg)
urFunctor_car A _ B _ := ofe_funUR (λ c, urFunctor_car (F c) A B);
urFunctor_map A1 _ A2 _ B1 _ B2 _ fg := ofe_funC_map (λ c, urFunctor_map (F c) fg)
|}.
Next Obligation.
intros C F A1 A2 B1 B2 n ?? g. by apply ofe_funC_map_ne=>?; apply urFunctor_ne.
intros C F A1 ? A2 ? B1 ? B2 ? n ?? g. by apply ofe_funC_map_ne=>?; apply urFunctor_ne.
Qed.
Next Obligation.
intros C F A B g; simpl. rewrite -{2}(ofe_fun_map_id g).
intros C F A ? B ? g; simpl. rewrite -{2}(ofe_fun_map_id g).
apply ofe_fun_map_ext=> y; apply urFunctor_id.
Qed.
Next Obligation.
intros C F A1 A2 A3 B1 B2 B3 f1 f2 f1' f2' g. rewrite /=-ofe_fun_map_compose.
intros C F A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f1 f2 f1' f2' g. rewrite /=-ofe_fun_map_compose.
apply ofe_fun_map_ext=>y; apply urFunctor_compose.
Qed.
Instance ofe_funURF_contractive {C} (F : C → urFunctor) :
(∀ c, urFunctorContractive (F c)) → urFunctorContractive (ofe_funURF F).
Proof.
intros ? A1 A2 B1 B2 n ?? g.
intros ? A1 ? A2 ? B1 ? B2 ? n ?? g.
by apply ofe_funC_map_ne=>c; apply urFunctor_contractive.
Qed.
theories/algebra/cofe_solver.v
View file @ 45d4e2a6
......@@ -4,8 +4,8 @@ Set Default Proof Using "Type".
Record solution (F : cFunctor) := Solution {
solution_car :> ofeT;
solution_cofe : Cofe solution_car;
solution_unfold : solution_car -n> F solution_car;
solution_fold : F solution_car -n> solution_car;
solution_unfold : solution_car -n> F solution_car _;
solution_fold : F solution_car _ -n> solution_car;
solution_fold_unfold X : solution_fold (solution_unfold X) ≡ X;
solution_unfold_fold X : solution_unfold (solution_fold X) ≡ X
}.
......@@ -14,21 +14,25 @@ Arguments solution_fold {_} _.
Existing Instance solution_cofe.
Module solver. Section solver.
Context (F : cFunctor) `{Fcontr : cFunctorContractive F}
`{Fcofe : ∀ T : ofeT, Cofe T → Cofe (F T)} `{Finh : Inhabited (F unitC)}.
Context (F : cFunctor) `{Fcontr : cFunctorContractive F}.
Context `{Fcofe : ∀ (T : ofeT) `{!Cofe T}, Cofe (F T _)}.
Context `{Finh : Inhabited (F unitC _)}.
Notation map := (cFunctor_map F).
Fixpoint A (k : nat) : ofeT :=
match k with 0 => unitC | S k => F (A k) end.
Local Instance: ∀ k, Cofe (A k).
Proof using Fcofe. induction k; apply _. Defined.
Fixpoint A' (k : nat) : { C : ofeT & Cofe C } :=
match k with
| 0 => existT (P:=Cofe) unitC _
| S k => existT (P:=Cofe) (F (projT1 (A' k)) (projT2 (A' k))) _
end.
Notation A k := (projT1 (A' k)).
Local Instance A_cofe k : Cofe (A k) := projT2 (A' k).
Fixpoint f (k : nat) : A k -n> A (S k) :=
match k with 0 => CofeMor (λ _, inhabitant) | S k => map (g k,f k) end
with g (k : nat) : A (S k) -n> A k :=
match k with 0 => CofeMor (λ _, ()) | S k => map (f k,g k) end.
Definition f_S k (x : A (S k)) : f (S k) x = map (g k,f k) x := eq_refl.
Definition g_S k (x : A (S (S k))) : g (S k) x = map (f k,g k) x := eq_refl.
Arguments A : simpl never.
Arguments f : simpl never.
Arguments g : simpl never.
......@@ -177,7 +181,7 @@ Proof.
- rewrite (ff_tower k (i - S k) X). by destruct (Nat.sub_add _ _ _).
Qed.
Program Definition unfold_chain (X : T) : chain (F T) :=
Program Definition unfold_chain (X : T) : chain (F T _) :=
{| chain_car n := map (project n,embed' n) (X (S n)) |}.
Next Obligation.
intros X n i Hi.
......@@ -187,14 +191,14 @@ Next Obligation.
rewrite f_S -cFunctor_compose.
by apply (contractive_ne map); split=> Y /=; rewrite ?g_tower ?embed_f.
Qed.
Definition unfold (X : T) : F T := compl (unfold_chain X).
Definition unfold (X : T) : F T _ := compl (unfold_chain X).
Instance unfold_ne : NonExpansive unfold.
Proof.
intros n X Y HXY. by rewrite /unfold (conv_compl n (unfold_chain X))
(conv_compl n (unfold_chain Y)) /= (HXY (S n)).
Qed.
Program Definition fold (X : F T) : T :=
Program Definition fold (X : F T _) : T :=
{| tower_car n := g n (map (embed' n,project n) X) |}.
Next Obligation.
intros X k. apply (_ : Proper ((≡) ==> (≡)) (g k)).
......
theories/algebra/csum.v
View file @ 45d4e2a6
......@@ -100,7 +100,7 @@ Global Instance csum_ofe_discrete :
OfeDiscrete A → OfeDiscrete B → OfeDiscrete csumC.
Proof. by inversion_clear 3; constructor; apply (discrete _). Qed.
Global Instance csum_leibniz :
LeibnizEquiv A → LeibnizEquiv B → LeibnizEquiv (csumC A B).
LeibnizEquiv A → LeibnizEquiv B → LeibnizEquiv csumC.
Proof. by destruct 3; f_equal; apply leibniz_equiv. Qed.
Global Instance Cinl_discrete a : Discrete a → Discrete (Cinl a).
......@@ -367,18 +367,18 @@ Proof.
Qed.
Program Definition csumRF (Fa Fb : rFunctor) : rFunctor := {|
rFunctor_car A B := csumR (rFunctor_car Fa A B) (rFunctor_car Fb A B);
rFunctor_map A1 A2 B1 B2 fg := csumC_map (rFunctor_map Fa fg) (rFunctor_map Fb fg)
rFunctor_car A _ B _ := csumR (rFunctor_car Fa A B) (rFunctor_car Fb A B);
rFunctor_map A1 _ A2 _ B1 _ B2 _ fg := csumC_map (rFunctor_map Fa fg) (rFunctor_map Fb fg)
|}.
Next Obligation.
by intros Fa Fb A1 A2 B1 B2 n f g Hfg; apply csumC_map_ne; try apply rFunctor_ne.
by intros Fa Fb A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply csumC_map_ne; try apply rFunctor_ne.
Qed.
Next Obligation.
intros Fa Fb A B x. rewrite /= -{2}(csum_map_id x).
intros Fa Fb A ? B ? x. rewrite /= -{2}(csum_map_id x).
apply csum_map_ext=>y; apply rFunctor_id.
Qed.
Next Obligation.
intros Fa Fb A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -csum_map_compose.
intros Fa Fb A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x. rewrite /= -csum_map_compose.
apply csum_map_ext=>y; apply rFunctor_compose.
Qed.
......@@ -386,5 +386,6 @@ Instance csumRF_contractive Fa Fb :
rFunctorContractive Fa → rFunctorContractive Fb →
rFunctorContractive (csumRF Fa Fb).
Proof.
by intros ?? A1 A2 B1 B2 n f g Hfg; apply csumC_map_ne; try apply rFunctor_contractive.
intros ?? A1 ? A2 ? B1 ? B2 ? n f g Hfg.
by apply csumC_map_ne; try apply rFunctor_contractive.
Qed.
theories/algebra/excl.v
View file @ 45d4e2a6
......@@ -150,23 +150,23 @@ Instance exclC_map_ne A B : NonExpansive (@exclC_map A B).
Proof. by intros n f f' Hf []; constructor; apply Hf. Qed.
Program Definition exclRF (F : cFunctor) : rFunctor := {|
rFunctor_car A B := (exclR (cFunctor_car F A B));
rFunctor_map A1 A2 B1 B2 fg := exclC_map (cFunctor_map F fg)
rFunctor_car A _ B _ := (exclR (cFunctor_car F A B));
rFunctor_map A1 _ A2 _ B1 _ B2 _ fg := exclC_map (cFunctor_map F fg)
|}.
Next Obligation.
intros F A1 A2 B1 B2 n x1 x2 ??. by apply exclC_map_ne, cFunctor_ne.
intros F A1 ? A2 ? B1 ? B2 ? n x1 x2 ??. by apply exclC_map_ne, cFunctor_ne.
Qed.
Next Obligation.
intros F A B x; simpl. rewrite -{2}(excl_map_id x).
intros F A ? B ? x; simpl. rewrite -{2}(excl_map_id x).
apply excl_map_ext=>y. by rewrite cFunctor_id.
Qed.
Next Obligation.
intros F A1 A2 A3 B1 B2 B3 f g f' g' x; simpl. rewrite -excl_map_compose.
intros F A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x; simpl. rewrite -excl_map_compose.
apply excl_map_ext=>y; apply cFunctor_compose.
Qed.
Instance exclRF_contractive F :
cFunctorContractive F → rFunctorContractive (exclRF F).
Proof.
intros A1 A2 B1 B2 n x1 x2 ??. by apply exclC_map_ne, cFunctor_contractive.
intros A1 ? A2 ? B1 ? B2 ? n x1 x2 ??. by apply exclC_map_ne, cFunctor_contractive.
Qed.
theories/algebra/gmap.v
View file @ 45d4e2a6
......@@ -558,43 +558,43 @@ Proof.
Qed.
Program Definition gmapCF K `{Countable K} (F : cFunctor) : cFunctor := {|
cFunctor_car A B := gmapC K (cFunctor_car F A B);
cFunctor_map A1 A2 B1 B2 fg := gmapC_map (cFunctor_map F fg)
cFunctor_car A _ B _ := gmapC K (cFunctor_car F A B);
cFunctor_map A1 _ A2 _ B1 _ B2 _ fg := gmapC_map (cFunctor_map F fg)
|}.
Next Obligation.
by intros K ?? F A1 A2 B1 B2 n f g Hfg; apply gmapC_map_ne, cFunctor_ne.
by intros K ?? F A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply gmapC_map_ne, cFunctor_ne.
Qed.
Next Obligation.
intros K ?? F A B x. rewrite /= -{2}(map_fmap_id x).
intros K ?? F A ? B ? x. rewrite /= -{2}(map_fmap_id x).
apply map_fmap_equiv_ext=>y ??; apply cFunctor_id.
Qed.
Next Obligation.
intros K ?? F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -map_fmap_compose.
intros K ?? F A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x. rewrite /= -map_fmap_compose.
apply map_fmap_equiv_ext=>y ??; apply cFunctor_compose.
Qed.
Instance gmapCF_contractive K `{Countable K} F :
cFunctorContractive F → cFunctorContractive (gmapCF K F).
Proof.
by intros ? A1 A2 B1 B2 n f g Hfg; apply gmapC_map_ne, cFunctor_contractive.