# Library iris.algebra.ofe

From iris.prelude Require Export prelude.
From iris.prelude Require Import options.
Local Set Primitive Projections.

This files defines (a shallow embedding of) the category of OFEs: Complete ordered families of equivalences. This is a cartesian closed category, and mathematically speaking, the entire development lives in this category. However, we will generally prefer to work with raw Coq functions plus some registered Proper instances for non-expansiveness. This makes writing such functions much easier. It turns out that it many cases, we do not even need non-expansiveness.
The tactic si_solver solves goals that are solely concerned with step-indices and their relations (i.e., 0, S n, n < m, and n m). Currently, this tactic is just an alias for lia. However, in the future, Iris will generalize over the type of step-indices, and this tactic will be able to solve step-indexing goals also in this generalized setting.
The tactic can be used as part of eauto by using the hint database si_solver.

Ltac si_solver := lia.

Create HintDb si_solver.
Global Hint Extern 1 ⇒ si_solver : si_solver.

Unbundled version
Class Dist A := dist : nat relation A.
Global Hint Mode Dist ! : typeclass_instances.
Global Instance: Params (@dist) 3 := {}.
Notation "x ≡{ n }≡ y" := (dist n x y)
(at level 70, n at next level, format "x ≡{ n }≡ y").
Notation "x ≡{ n }@{ A }≡ y" := (dist (A:=A) n x y)
(at level 70, n at next level, only parsing).
Notation "(≡{ n }≡)" := (dist n) (only parsing).
Notation "(≡{ n }@{ A }≡)" := (dist (A:=A) n) (only parsing).
Notation "( x ≡{ n }≡.)" := (dist n x) (only parsing).
Notation "(.≡{ n }≡ y )" := (λ x, x ≡{n}≡ y) (only parsing).

Global Hint Extern 0 (_ ≡{_}≡ _) ⇒ reflexivity : core.
Global Hint Extern 0 (_ ≡{_}≡ _) ⇒ symmetry; assumption : core.
Notation NonExpansive f := ( n, Proper (dist n ==> dist n) f).
Notation NonExpansive2 f := ( n, Proper (dist n ==> dist n ==> dist n) f).
Notation NonExpansive3 f :=
( n, Proper (dist n ==> dist n ==> dist n ==> dist n) f).
Notation NonExpansive4 f :=
( n, Proper (dist n ==> dist n ==> dist n ==> dist n ==> dist n) f).

Tactic Notation "ofe_subst" ident(x) :=
repeat match goal with
| _progress simplify_eq/=
| H:@dist ?A ?d ?n x _ |- _setoid_subst_aux (@dist A d n) x
| H:@dist ?A ?d ?n _ x |- _symmetry in H;setoid_subst_aux (@dist A d n) x
end.
Tactic Notation "ofe_subst" :=
repeat match goal with
| _progress simplify_eq/=
| H:@dist ?A ?d ?n ?x _ |- _setoid_subst_aux (@dist A d n) x
| H:@dist ?A ?d ?n _ ?x |- _symmetry in H;setoid_subst_aux (@dist A d n) x
end.

Record OfeMixin A `{Equiv A, Dist A} := {
mixin_equiv_dist (x y : A) : x y n, x ≡{n}≡ y;
mixin_dist_equivalence n : Equivalence (@dist A _ n);
mixin_dist_lt n m (x y : A) : x ≡{n}≡ y m < n x ≡{m}≡ y;
}.

Bundled version
Structure ofe := Ofe {
ofe_car :> Type;
ofe_equiv : Equiv ofe_car;
ofe_dist : Dist ofe_car;
ofe_mixin : OfeMixin ofe_car
}.
Global Arguments Ofe _ {_ _} _.
Add Printing Constructor ofe.
Global Hint Extern 0 (Equiv _) ⇒ refine (ofe_equiv _); shelve : typeclass_instances.
Global Hint Extern 0 (Dist _) ⇒ refine (ofe_dist _); shelve : typeclass_instances.
Global Arguments ofe_car : simpl never.
Global Arguments ofe_equiv : simpl never.
Global Arguments ofe_dist : simpl never.
Global Arguments ofe_mixin : simpl never.

When declaring instances of subclasses of OFE (like CMRAs and unital CMRAs) we need Coq to *infer* the canonical OFE instance of a given type and take the mixin out of it. This makes sure we do not use two different OFE instances in different places (see for example the constructors Cmra and Ucmra in the file cmra.v.)
In order to infer the OFE instance, we use the definition ofe_mixin_of' which is inspired by the clone trick in ssreflect. It works as follows, when type checking A ?Ac id Coq faces a unification problem:
ofe_car ?Ac ~ A
which will resolve ?Ac to the canonical OFE instance corresponding to A. The definition A ?Ac id will then provide the corresponding mixin. Note that type checking of ofe_mixin_of' A id will fail when A does not have a canonical OFE instance.
The notation ofe_mixin_of A that we define on top of ofe_mixin_of' A id hides the id and normalizes the mixin to head normal form. The latter is to ensure that we do not end up with redundant canonical projections to the mixin, i.e. them all being of the shape ofe_mixin_of' A id.
Definition ofe_mixin_of' A {Ac : ofe} (f : Ac A) : OfeMixin Ac := ofe_mixin Ac.
Notation ofe_mixin_of A :=
ltac:(let H := eval hnf in (ofe_mixin_of' A id) in exact H) (only parsing).

Lifting properties from the mixin
Section ofe_mixin.
Context {A : ofe}.
Implicit Types x y : A.
Lemma equiv_dist x y : x y n, x ≡{n}≡ y.
Proof. apply (mixin_equiv_dist _ (ofe_mixin A)). Qed.
Global Instance dist_equivalence n : Equivalence (@dist A _ n).
Proof. apply (mixin_dist_equivalence _ (ofe_mixin A)). Qed.
Lemma dist_lt n m x y : x ≡{n}≡ y m < n x ≡{m}≡ y.
Proof. apply (mixin_dist_lt _ (ofe_mixin A)). Qed.
End ofe_mixin.

Global Hint Extern 1 (_ ≡{_}≡ _) ⇒ apply equiv_dist; assumption : core.

Discrete OFEs and discrete OFE elements
Class Discrete {A : ofe} (x : A) := discrete_0 y : x ≡{0}≡ y x y.
Global Arguments discrete_0 {_} _ {_} _ _.
Global Hint Mode Discrete + ! : typeclass_instances.
Global Instance: Params (@Discrete) 1 := {}.

Class OfeDiscrete (A : ofe) := #[global] ofe_discrete_discrete (x : A) :: Discrete x.
Global Hint Mode OfeDiscrete ! : typeclass_instances.

OFEs with a completion
Record chain (A : ofe) := {
chain_car :> nat A;
chain_cauchy n i : n i chain_car i ≡{n}≡ chain_car n
}.
Global Arguments chain_car {_} _ _.
Global Arguments chain_cauchy {_} _ _ _ _.

Program Definition chain_map {A B : ofe} (f : A B)
`{!NonExpansive f} (c : chain A) : chain B :=
{| chain_car n := f (c n) |}.
Next Obligation. by intros A B f Hf c n i ?; apply Hf, chain_cauchy. Qed.

Notation Compl A := (chain A%type A).
Class Cofe (A : ofe) := {
compl : Compl A;
conv_compl n c : compl c ≡{n}≡ c n;
}.
Global Arguments compl : simpl never.
Global Hint Mode Cofe ! : typeclass_instances.

Lemma compl_chain_map `{!Cofe A, !Cofe B} (f : A B) c `(!NonExpansive f) :
compl (chain_map f c) f (compl c).
Proof. apply equiv_distn. by rewrite !conv_compl. Qed.

Program Definition chain_const {A : ofe} (a : A) : chain A :=
{| chain_car n := a |}.
Next Obligation. by intros A a n i _. Qed.

Lemma compl_chain_const {A : ofe} `{!Cofe A} (a : A) :
compl (chain_const a) a.
Proof. apply equiv_distn. by rewrite conv_compl. Qed.

General properties
Section ofe.
Context {A : ofe}.
Implicit Types x y : A.
Global Instance ofe_equivalence : Equivalence ((≡) : relation A).
Proof.
split.
- by intros x; rewrite equiv_dist.
- by intros x y; rewrite !equiv_dist.
- by intros x y z; rewrite !equiv_dist; intros; trans y.
Qed.
Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n).
Proof.
intros x1 x2 ? y1 y2 ?; split; intros.
- by trans x1; [|trans y1].
- by trans x2; [|trans y2].
Qed.
Global Instance dist_proper n : Proper ((≡) ==> (≡) ==> iff) (@dist A _ n).
Proof.
by movex1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n).
Qed.
Global Instance dist_proper_2 n x : Proper ((≡) ==> iff) (dist n x).
Proof. by apply dist_proper. Qed.
Global Instance Discrete_proper : Proper ((≡) ==> iff) (@Discrete A).
Proof. intros x y Hxy. rewrite /Discrete. by setoid_rewrite Hxy. Qed.

Lemma dist_le n n' x y : x ≡{n}≡ y n' n x ≡{n'}≡ y.
Proof. intros ? [Hm | ->]%Nat.lt_eq_cases; [by eapply dist_lt | auto]. Qed.
Lemma dist_le' n n' x y : n' n x ≡{n}≡ y x ≡{n'}≡ y.
Proof. eauto using dist_le. Qed.
Lemma dist_S n x y : x ≡{S n}≡ y x ≡{n}≡ y.
Proof. eauto using dist_le. Qed.
ne_proper and ne_proper_2 are not instances to improve efficiency of type class search during setoid rewriting. Local Instances of NonExpansive{,2} are hence accompanied by instances of Proper built using these lemmas.
Lemma ne_proper {B : ofe} (f : A B) `{!NonExpansive f} :
Proper ((≡) ==> (≡)) f.
Proof. by intros x1 x2; rewrite !equiv_dist; intros Hx n; rewrite (Hx n). Qed.
Lemma ne_proper_2 {B C : ofe} (f : A B C) `{!NonExpansive2 f} :
Proper ((≡) ==> (≡) ==> (≡)) f.
Proof.
unfold Proper, respectful; setoid_rewrite equiv_dist.
by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n).
Qed.

Lemma conv_compl' `{!Cofe A} n (c : chain A) : compl c ≡{n}≡ c (S n).
Proof.
transitivity (c n); first by apply conv_compl. symmetry.
apply chain_cauchy. lia.
Qed.

Lemma discrete_iff n (x : A) `{!Discrete x} y : x y x ≡{n}≡ y.
Proof.
split; intros; auto. apply (discrete_0 _), dist_le with n; auto with lia.
Qed.
Lemma discrete_iff_0 n (x : A) `{!Discrete x} y : x ≡{0}≡ y x ≡{n}≡ y.
Proof. by rewrite -!discrete_iff. Qed.
Lemma discrete n (x : A) `{!Discrete x} y : x ≡{n}≡ y x y.
Proof. intros. eapply discrete_iff; done. Qed.

Global Instance ofe_discrete_subrelation `{!OfeDiscrete A} n :
@SolveProperSubrelation A (dist n) (≡).
Proof. intros ???. apply: discrete. done. Qed.
Global Instance ofe_leibniz_subrelation `{!OfeDiscrete A, !LeibnizEquiv A} n :
@SolveProperSubrelation A (dist n) (=).
Proof. intros ?? EQ. unfold_leibniz. apply (is_solve_proper_subrelation EQ). Qed.
End ofe.

Contractive functions Defined as a record to avoid eager unfolding.
Record dist_later `{!Dist A} n (x y : A) : Prop :=
{ dist_later_lt : m, m < n x ≡{m}≡ y }.

Section dist_later.
Context {A : ofe}.
Implicit Types x y : A.

Global Instance dist_later_equivalence n : Equivalence (@dist_later A _ n).
Proof.
split.
- intros ?; by split.
- intros ?? [Hlater]; split; intros ??; by rewrite Hlater.
- intros ??? [Hlater1] [Hlater2]; split; intros ??; by rewrite Hlater1 ?Hlater2.
Qed.

Lemma dist_dist_later n x y : dist n x y dist_later n x y.
Proof. intros. split; eauto using dist_le. Qed.

Lemma dist_later_dist_lt n m x y : m < n dist_later n x y dist m x y.
Proof. intros ? []; eauto. Qed.

Lemma dist_later_0 x y : dist_later 0 x y.
Proof. split; intros ? []%Nat.nlt_0_r. Qed.

Lemma dist_later_S n x y: x ≡{n}≡ y dist_later (S n) x y.
Proof.
split.
- intros Hn; split; intros m Hm. eapply dist_le; first done. lia.
- intros Hdist. apply Hdist. lia.
Qed.
End dist_later.

Lemma ne_dist_later {A B : ofe} (f : A B) :
NonExpansive f n, Proper (dist_later n ==> dist_later n) f.
Proof. intros Hf ??? Hlater; split; intros ??; by eapply Hf, Hlater. Qed.

We define dist_later_fin, an equivalent (see dist_later_fin_iff) version of dist_later that uses a match on the step-index instead of the quantification over smaller step-indicies. The definition of dist_later_fin matches how dist_later used to be defined (i.e., with a match on the step-index), so dist_later_fin simplifies adapting existing Iris developments that used to rely on the reduction behavior of dist_later.
The "fin" indicates that when, in the future, the step-index is abstracted away, this equivalence will only hold for finite step-indices (as in, ordinals without "limit" steps such as natural numbers).
Definition dist_later_fin {A : ofe} (n : nat) (x y : A) :=
match n with 0 ⇒ True | S nx ≡{n}≡ y end.

Lemma dist_later_fin_iff {A : ofe} (n : nat) (x y : A):
dist_later n x y dist_later_fin n x y.
Proof.
destruct n; unfold dist_later_fin; first by split; eauto using dist_later_0.
by rewrite dist_later_S.
Qed.

Notation Contractive f := ( n, Proper (dist_later n ==> dist n) f).

Global Instance const_contractive {A B : ofe} (x : A) : Contractive (@const A B x).
Proof. by intros n y1 y2. Qed.

Section contractive.
Local Set Default Proof Using "Type*".
Context {A B : ofe} (f : A B) `{!Contractive f}.
Implicit Types x y : A.

Lemma contractive_0 x y : f x ≡{0}≡ f y.
Proof. by apply (_ : Contractive f), dist_later_0. Qed.
Lemma contractive_dist_later_dist n x y : dist_later n x y f x ≡{n}≡ f y.
Proof. intros. by apply (_ : Contractive f). Qed.
Lemma contractive_S n x y : x ≡{n}≡ y f x ≡{S n}≡ f y.
Proof. intros. by apply contractive_dist_later_dist, dist_later_S. Qed.

Global Instance contractive_ne : NonExpansive f | 100.
Proof. intros n x y ?. eapply contractive_dist_later_dist. by apply dist_dist_later. Qed.
Global Instance contractive_proper : Proper ((≡) ==> (≡)) f | 100.
Proof. apply (ne_proper _). Qed.
End contractive.

Lemma dist_pointwise_lt {A} {B: ofe} n m (f g: A B):
m < n
pointwise_relation A (dist_later n) f g
pointwise_relation A (dist m) f g.
Proof. intros Hlt Hp a. by apply Hp. Qed.

The tactic f_contractive can be used to prove contractiveness or non-expansiveness of a function f. Inside of the proof of contractiveness/non-expansiveness, if the current goal is g x1 ... xn ≡{i}≡ g y1 ... yn for a contractive function g (that is used inside of the body of f), then the tactic will try to find a suitable Contractive instance for g and apply it. Currently, the tactic only supports one (i.e., n = 1) and two (i.e., n = 2) arguments. As a result of applying the Contractive instance for g, one of the goals will be dist_later i xi yi and the tactic will try to simplify or solve the goal. By simplify we mean that it will turn hypotheses dist_later into dist.
For backwards compatibility, we also define the tactic f_contractive_fin that works with an earlier definition of dist_later now called dist_later_fin. The new version of f_contractive is future proof with respect to generalizing the type of step-indices, while the old tactic relies crucially on the step-indices being nat and the reduction behavior of dist_later. The tactic f_contractive_fin simplifies backwards compatibility of existing Iris developments (e.g., RustBelt), that define custom notions of dist and dist_later but should be avoided if possible.
The tactics f_contractive and f_contractive_fin are implemented using
1. f_contractive_prepare which looks up a Contractive looks at which function is being applied on both sides of a dist, looks up the Contractive instance (or the equivalent for two arguments) and applies it. 2. dist_later_intro and dist_later_fin_intro which introduces the resulting goals with dist_later n x y/dist_later_fin n x y. The tactic dist_later_intro works with the normal definition of dist_later and is future compatible with generalizing the step-index beyond natural numbers. The tactic dist_later_fin_intro is a special case which only works for natural numbers as step-indicies. It changes dist_later to dist_later_fin, which only makes sense on natural numbers. We keep dist_later_fin_intro around for backwards compatibility.
Ltac f_contractive_prepare :=
match goal with
| |- ?f _ ≡{_}≡ ?f _simple apply (_ : Proper (dist_later _ ==> dist _) f)
| |- ?f _ _ ≡{_}≡ ?f _ _simple apply (_ : Proper (dist_later _ ==> _ ==> dist _) f)
| |- ?f _ _ ≡{_}≡ ?f _ _simple apply (_ : Proper (_ ==> dist_later _ ==> dist _) f)
end.

For the goal dist_later n x y, the tactic dist_later_intro as m Hm introduces a smaller step-index Hm : m < n and tries to lower assumptions in the context to m where possible. The arguments m and Hm can be omitted, in which case a fresh identifier is used.
Tactic Notation "dist_later_intro" "as" ident(idxName) ident(ltName) :=
match goal with
| |- dist_later ?n ?x ?y
constructor; intros idxName ltName;
repeat match goal with
| H: dist_later n _ _ |- _destruct H as [H]; specialize (H idxName ltName) as H
| H: pointwise_relation _ (dist_later n) _ _ |- _
apply (dist_pointwise_lt _ idxName _ _ ltName) in H
end
end.
Tactic Notation "dist_later_intro" :=
let m := fresh "m" in
let Hlt := fresh "Hlt" in
dist_later_intro as m Hlt.

For the goal dist_later n x y, the tactic dist_later_fin_intro changes the goal to dist_later_fin and takes care of the case where n=0, such that we are only left with the case where n = S n' for some n'. Changing dist_later to dist_later_fin enables reduction and thus works better with custom versions of dist as used e.g. by LambdaRust.
Ltac dist_later_fin_intro :=
match goal with
| |- @dist_later ?A _ ?n ?x ?y
apply dist_later_fin_iff;
destruct n as [|n]; [exact I|change (@dist A _ n x y)]
end.

We combine f_contractive_prepare and dist_later_intro into the f_contractive tactic.
For all the goals not solved by dist_later_intro (i.e., the ones that are not dist_later n x y), we try reflexivity. Since reflexivity can be very expensive when unification fails, we use fast_reflexivity.

Tactic Notation "f_contractive" "as" ident(idxName) ident(ltName) :=
f_contractive_prepare;
try dist_later_intro as idxName ltName;
try fast_reflexivity.

Tactic Notation "f_contractive" :=
let m := fresh "m" in
let Hlt := fresh "Hlt" in
f_contractive as m Hlt.

Tactic Notation "f_contractive_fin" :=
f_contractive_prepare;
try dist_later_fin_intro;
try fast_reflexivity.

Ltac solve_contractive :=
solve_proper_core ltac:(fun _first [f_contractive | f_equiv]).

Limit preserving predicates
Class LimitPreserving `{!Cofe A} (P : A Prop) : Prop :=
limit_preserving (c : chain A) : ( n, P (c n)) P (compl c).
Global Hint Mode LimitPreserving + + ! : typeclass_instances.

Section limit_preserving.
Context {A : ofe} `{!Cofe A}.

Lemma limit_preserving_ext (P Q : A Prop) :
( x, P x Q x) LimitPreserving P LimitPreserving Q.
Proof. intros HP Hlimit c ?. apply HP, Hlimitn; by apply HP. Qed.

Global Instance limit_preserving_const (P : Prop) : LimitPreserving (λ _ : A, P).
Proof. intros c HP. apply (HP 0). Qed.

Lemma limit_preserving_discrete (P : A Prop) :
Proper (dist 0 ==> impl) P LimitPreserving P.
Proof. intros PH c Hc. by rewrite (conv_compl 0). Qed.

Lemma limit_preserving_and (P1 P2 : A Prop) :
LimitPreserving P1
LimitPreserving P2
LimitPreserving (λ x, P1 x P2 x).
Proof.
intros Hlim1 Hlim2 c Hc.
split.
- apply Hlim1, Hc.
- apply Hlim2, Hc.
Qed.

Lemma limit_preserving_impl (P1 P2 : A Prop) :
Proper (dist 0 ==> impl) P1
LimitPreserving P2
LimitPreserving (λ x, P1 x P2 x).
Proof.
intros Hlim1 Hlim2 c Hc HP1. apply Hlim2n; apply Hc.
eapply Hlim1, HP1. apply dist_le with n; last lia. apply (conv_compl n).
Qed.

This is strictly weaker than the _impl variant, but sometimes automation is better at proving Proper for iff than for impl.
Lemma limit_preserving_impl' (P1 P2 : A Prop) :
Proper (dist 0 ==> iff) P1
LimitPreserving P2
LimitPreserving (λ x, P1 x P2 x).
Proof.
intros HP1. apply limit_preserving_impl. intros ???.
apply iff_impl_subrelation. eapply HP1. done.
Qed.

Lemma limit_preserving_forall {B} (P : B A Prop) :
( y, LimitPreserving (P y))
LimitPreserving (λ x, y, P y x).
Proof. intros Hlim c Hc y. by apply Hlim. Qed.

Lemma limit_preserving_equiv `{!Cofe B} (f g : A B) :
NonExpansive f NonExpansive g LimitPreserving (λ x, f x g x).
Proof.
intros Hf Hg c Hc. apply equiv_distn.
by rewrite -!compl_chain_map !conv_compl /= Hc.
Qed.
End limit_preserving.

Fixpoint
Program Definition fixpoint_chain {A : ofe} `{Inhabited A} (f : A A)
`{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}.
Next Obligation.
intros A ? f ? n.
induction n as [|n IH]=> -[|i] //= ?; try lia.
- apply (contractive_0 f).
- apply (contractive_S f), IH; auto with lia.
Qed.

Local Program Definition fixpoint_def `{Cofe A, Inhabited A} (f : A A)
`{!Contractive f} : A := compl (fixpoint_chain f).
Local Definition fixpoint_aux : seal (@fixpoint_def). Proof. by eexists. Qed.
Definition fixpoint := fixpoint_aux.(unseal).
Global Arguments fixpoint {A _ _} f {_}.
Local Definition fixpoint_unseal :
@fixpoint = @fixpoint_def := fixpoint_aux.(seal_eq).

Section fixpoint.
Context `{!Cofe A, !Inhabited A} (f : A A) `{!Contractive f}.

This lemma does not work well with rewrite; we usually define a specific unfolding lemma for each fixpoint and then apply fixpoint_unfold in the proof of that unfolding lemma.
Lemma fixpoint_unfold : fixpoint f f (fixpoint f).
Proof.
apply equiv_distn.
rewrite fixpoint_unseal /fixpoint_def (conv_compl n (fixpoint_chain f)) //.
induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S.
Qed.

Lemma fixpoint_unique (x : A) : x f x x fixpoint f.
Proof.
rewrite !equiv_distHx n. induction n as [|n IH]; simpl in ×.
- rewrite Hx fixpoint_unfold; eauto using contractive_0.
- rewrite Hx fixpoint_unfold. eauto using contractive_S.
Qed.

Lemma fixpoint_ne (g : A A) `{!Contractive g} n :
( z, f z ≡{n}≡ g z) fixpoint f ≡{n}≡ fixpoint g.
Proof.
intros Hfg. rewrite fixpoint_unseal /fixpoint_def
(conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=.
induction n as [|n IH]; simpl in *; [by rewrite !Hfg|].
rewrite Hfg; apply contractive_S, IH; eauto using dist_le with si_solver.
Qed.
Lemma fixpoint_proper (g : A A) `{!Contractive g} :
( x, f x g x) fixpoint f fixpoint g.
Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed.

Lemma fixpoint_ind (P : A Prop) :
Proper ((≡) ==> impl) P
( x, P x) ( x, P x P (f x))
LimitPreserving P
P (fixpoint f).
Proof.
intros ? [x Hx] Hincr Hlim. set (chcar i := Nat.iter (S i) f x).
assert (Hcauch : n i : nat, n i chcar i ≡{n}≡ chcar n).
{ intros n. rewrite /chcar. induction n as [|n IH]=> -[|i] //=;
eauto using contractive_0, contractive_S with si_solver. }
set (fp2 := compl {| chain_cauchy := Hcauch |}).
assert (f fp2 fp2).
{ apply equiv_distn. rewrite /fp2 (conv_compl n) /= /chcar.
induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S. }
rewrite -(fixpoint_unique fp2) //.
apply Hlimn /=. by apply Nat.iter_ind.
Qed.
End fixpoint.

Fixpoint of f when f^k is contractive.
Definition fixpointK {A : ofe} `{!Cofe A, !Inhabited A} k (f : A A)
`{!Contractive (Nat.iter k f)} := fixpoint (Nat.iter k f).

Section fixpointK.
Local Set Default Proof Using "Type*".
Context {A : ofe} `{!Cofe A, !Inhabited A} (f : A A) (k : nat).
Context {f_contractive : Contractive (Nat.iter k f)} {f_ne : NonExpansive f}.

Let f_proper : Proper ((≡) ==> (≡)) f := ne_proper f.
Local Existing Instance f_proper.

Lemma fixpointK_unfold : fixpointK k f f (fixpointK k f).
Proof.
symmetry. rewrite /fixpointK. apply fixpoint_unique.
by rewrite -Nat.iter_succ_r Nat.iter_succ -fixpoint_unfold.
Qed.

Lemma fixpointK_unique (x : A) : x f x x fixpointK k f.
Proof.
intros Hf. apply fixpoint_unique. clear f_contractive.
induction k as [|k' IH]=> //=. by rewrite -IH.
Qed.

Section fixpointK_ne.
Context (g : A A) `{g_contractive : !Contractive (Nat.iter k g)}.
Context {g_ne : NonExpansive g}.

Lemma fixpointK_ne n : ( z, f z ≡{n}≡ g z) fixpointK k f ≡{n}≡ fixpointK k g.
Proof.
rewrite /fixpointKHfg /=. apply fixpoint_nez.
clear f_contractive g_contractive.
induction k as [|k' IH]=> //=. by rewrite IH Hfg.
Qed.

Lemma fixpointK_proper : ( z, f z g z) fixpointK k f fixpointK k g.
Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpointK_ne. Qed.
End fixpointK_ne.

Lemma fixpointK_ind (P : A Prop) :
Proper ((≡) ==> impl) P
( x, P x) ( x, P x P (f x))
LimitPreserving P
P (fixpointK k f).
Proof.
intros. rewrite /fixpointK. apply fixpoint_ind; eauto.
intros; apply Nat.iter_ind; auto.
Qed.
End fixpointK.

Mutual fixpoints
Section fixpointAB.
Context {A B : ofe} `{!Cofe A, !Cofe B, !Inhabited A, !Inhabited B}.
Context (fA : A B A).
Context (fB : A B B).
Context {fA_contractive : n, Proper (dist_later n ==> dist n ==> dist n) fA}.
Context {fB_contractive : n, Proper (dist_later n ==> dist_later n ==> dist n) fB}.

Local Definition fixpoint_AB (x : A) : B := fixpoint (fB x).
Local Instance fixpoint_AB_contractive : Contractive fixpoint_AB.
Proof.
intros n x x' Hx; rewrite /fixpoint_AB.
apply fixpoint_ney. by f_contractive.
Qed.

Local Definition fixpoint_AA (x : A) : A := fA x (fixpoint_AB x).
Local Instance fixpoint_AA_contractive : Contractive fixpoint_AA.
Proof using fA_contractive. solve_contractive. Qed.

Definition fixpoint_A : A := fixpoint fixpoint_AA.
Definition fixpoint_B : B := fixpoint_AB fixpoint_A.

Lemma fixpoint_A_unfold : fA fixpoint_A fixpoint_B fixpoint_A.
Proof. by rewrite {2}/fixpoint_A (fixpoint_unfold _). Qed.
Lemma fixpoint_B_unfold : fB fixpoint_A fixpoint_B fixpoint_B.
Proof. by rewrite {2}/fixpoint_B /fixpoint_AB (fixpoint_unfold _). Qed.

Local Instance: Proper ((≡) ==> (≡) ==> (≡)) fA.
Proof using fA_contractive.
apply ne_proper_2n x x' ? y y' ?. f_contractive;
eauto using dist_le with si_solver.
Qed.
Local Instance: Proper ((≡) ==> (≡) ==> (≡)) fB.
Proof using fB_contractive.
apply ne_proper_2n x x' ? y y' ?. f_contractive;
eauto using dist_le with si_solver.
Qed.

Lemma fixpoint_A_unique p q : fA p q p fB p q q p fixpoint_A.
Proof.
intros HfA HfB. rewrite -HfA. apply fixpoint_unique. rewrite /fixpoint_AA.
f_equiv⇒ //. apply fixpoint_unique. by rewrite HfA HfB.
Qed.
Lemma fixpoint_B_unique p q : fA p q p fB p q q q fixpoint_B.
Proof. intros. apply fixpoint_unique. by rewrite -fixpoint_A_unique. Qed.
End fixpointAB.

Section fixpointAB_ne.
Context {A B : ofe} `{!Cofe A, !Cofe B, !Inhabited A, !Inhabited B}.
Context (fA fA' : A B A).
Context (fB fB' : A B B).
Context `{ n, Proper (dist_later n ==> dist n ==> dist n) fA}.
Context `{ n, Proper (dist_later n ==> dist n ==> dist n) fA'}.
Context `{ n, Proper (dist_later n ==> dist_later n ==> dist n) fB}.
Context `{ n, Proper (dist_later n ==> dist_later n ==> dist n) fB'}.

Lemma fixpoint_A_ne n :
( x y, fA x y ≡{n}≡ fA' x y) ( x y, fB x y ≡{n}≡ fB' x y)
fixpoint_A fA fB ≡{n}≡ fixpoint_A fA' fB'.
Proof.
intros HfA HfB. apply fixpoint_nez.
rewrite /fixpoint_AA /fixpoint_AB HfA. f_equiv. by apply fixpoint_ne.
Qed.
Lemma fixpoint_B_ne n :
( x y, fA x y ≡{n}≡ fA' x y) ( x y, fB x y ≡{n}≡ fB' x y)
fixpoint_B fA fB ≡{n}≡ fixpoint_B fA' fB'.
Proof.
intros HfA HfB. apply fixpoint_nez. rewrite HfB. f_contractive.
apply fixpoint_A_ne; eauto using dist_le with si_solver.
Qed.

Lemma fixpoint_A_proper :
( x y, fA x y fA' x y) ( x y, fB x y fB' x y)
fixpoint_A fA fB fixpoint_A fA' fB'.
Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_A_ne. Qed.
Lemma fixpoint_B_proper :
( x y, fA x y fA' x y) ( x y, fB x y fB' x y)
fixpoint_B fA fB fixpoint_B fA' fB'.
Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_B_ne. Qed.
End fixpointAB_ne.

Non-expansive function space
Record ofe_mor (A B : ofe) : Type := OfeMor {
ofe_mor_car :> A B;
ofe_mor_ne : NonExpansive ofe_mor_car
}.
Global Arguments OfeMor {_ _} _ {_}.
Add Printing Constructor ofe_mor.
Global Existing Instance ofe_mor_ne.

Notation "'λne' x .. y , t" :=
(@OfeMor _ _ (λ x, .. (@OfeMor _ _ (λ y, t) _) ..) _)
(at level 200, x binder, y binder, right associativity).

Section ofe_mor.
Context {A B : ofe}.
Global Instance ofe_mor_proper (f : ofe_mor A B) : Proper ((≡) ==> (≡)) f.
Proof. apply ne_proper, ofe_mor_ne. Qed.
Local Instance ofe_mor_equiv : Equiv (ofe_mor A B) := λ f g, x, f x g x.
Local Instance ofe_mor_dist : Dist (ofe_mor A B) := λ n f g, x, f x ≡{n}≡ g x.
Definition ofe_mor_ofe_mixin : OfeMixin (ofe_mor A B).
Proof.
split.
- intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
intros Hfg k; apply equiv_distn; apply Hfg.
- intros n; split.
+ by intros f x.
+ by intros f g ? x.
+ by intros f g h ?? x; trans (g x).
- intros n m f g ? x ?; eauto using dist_le with si_solver.
Qed.
Canonical Structure ofe_morO := Ofe (ofe_mor A B) ofe_mor_ofe_mixin.

Program Definition ofe_mor_chain (c : chain ofe_morO)
(x : A) : chain B := {| chain_car n := c n x |}.
Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed.
Program Definition ofe_mor_compl `{!Cofe B} : Compl ofe_morO := λ c,
{| ofe_mor_car x := compl (ofe_mor_chain c x) |}.
Next Obligation.
intros ? c n x y Hx. by rewrite (conv_compl n (ofe_mor_chain c x))
(conv_compl n (ofe_mor_chain c y)) /= Hx.
Qed.
Global Program Instance ofe_mor_cofe `{!Cofe B} : Cofe ofe_morO :=
{| compl := ofe_mor_compl |}.
Next Obligation.
intros ? n c x; simpl.
by rewrite (conv_compl n (ofe_mor_chain c x)) /=.
Qed.

Global Instance ofe_mor_car_ne :
NonExpansive2 (@ofe_mor_car A B).
Proof. intros n f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed.
Global Instance ofe_mor_car_proper :
Proper ((≡) ==> (≡) ==> (≡)) (@ofe_mor_car A B) := ne_proper_2 _.
Lemma ofe_mor_ext (f g : ofe_mor A B) : f g x, f x g x.
Proof. done. Qed.
End ofe_mor.

Global Arguments ofe_morO : clear implicits.
Notation "A -n> B" :=
(ofe_morO A B) (at level 99, B at level 200, right associativity).
Global Instance ofe_mor_inhabited {A B : ofe} `{Inhabited B} :
Inhabited (A -n> B) := populate (λne _, inhabitant).

Identity and composition and constant function
Definition cid {A} : A -n> A := OfeMor id.
Global Instance: Params (@cid) 1 := {}.
Definition cconst {A B : ofe} (x : B) : A -n> B := OfeMor (const x).
Global Instance: Params (@cconst) 2 := {}.

Definition ccompose {A B C}
(f : B -n> C) (g : A -n> B) : A -n> C := OfeMor (f g).
Global Instance: Params (@ccompose) 3 := {}.
Infix "◎" := ccompose (at level 40, left associativity).
Global Instance ccompose_ne {A B C} :
NonExpansive2 (@ccompose A B C).
Proof. intros n ?? Hf g1 g2 Hg x. rewrite /= (Hg x) (Hf (g2 x)) //. Qed.
Global Instance ccompose_proper {A B C} :
Proper ((≡) ==> (≡) ==> (≡)) (@ccompose A B C).
Proof. apply ne_proper_2; apply _. Qed.

Definition ofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B')
(h : A -n> B) : A' -n> B' := g h f.
Global Instance ofe_mor_map_ne {A A' B B'} :
NonExpansive3 (@ofe_mor_map A A' B B').
Proof. intros n ??? ??? ???. by repeat apply ccompose_ne. Qed.

Definition ofe_morO_map {A A' B B'} (f : A' -n> A) (g : B -n> B') :
(A -n> B) -n> (A' -n> B') := OfeMor (ofe_mor_map f g).
Global Instance ofe_morO_map_ne {A A' B B'} :
NonExpansive2 (@ofe_morO_map A A' B B').
Proof.
intros n f f' Hf g g' Hg ?. rewrite /= /ofe_mor_map.
by repeat apply ccompose_ne.
Qed.

# Unit type

Section unit.
Local Instance unit_dist : Dist unit := λ _ _ _, True.
Definition unit_ofe_mixin : OfeMixin unit.
Proof. by repeat split; try 0. Qed.
Canonical Structure unitO : ofe := Ofe unit unit_ofe_mixin.

Global Program Instance unit_cofe : Cofe unitO := { compl x := () }.
Next Obligation. by repeat split; try 0. Qed.

Global Instance unit_ofe_discrete : OfeDiscrete unitO.
Proof. done. Qed.
End unit.

# Empty type

Section empty.
Local Instance Empty_set_dist : Dist Empty_set := λ _ _ _, True.
Definition Empty_set_ofe_mixin : OfeMixin Empty_set.
Proof. by repeat split; try 0. Qed.
Canonical Structure Empty_setO : ofe := Ofe Empty_set Empty_set_ofe_mixin.

Global Program Instance Empty_set_cofe : Cofe Empty_setO := { compl x := x 0 }.
Next Obligation. by repeat split; try 0. Qed.

Global Instance Empty_set_ofe_discrete : OfeDiscrete Empty_setO.
Proof. done. Qed.
End empty.

# Product type

Section product.
Context {A B : ofe}.

Local Instance prod_dist : Dist (A × B) := λ n, prod_relation (dist n) (dist n).

Definition prod_ofe_mixin : OfeMixin (A × B).
Proof.
split.
- intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation.
rewrite !equiv_dist; naive_solver.
- apply _.
- by intros n m [x1 y1] [x2 y2] [??] ?; split;
eauto using dist_le with si_solver.
Qed.
Canonical Structure prodO : ofe := Ofe (A × B) prod_ofe_mixin.

Global Program Instance prod_cofe `{Cofe A, Cofe B} : Cofe prodO :=
{ compl c := (compl (chain_map fst c), compl (chain_map snd c)) }.
Next Obligation.
intros ?? n c; split.
- apply (conv_compl n (chain_map fst c)).
- apply (conv_compl n (chain_map snd c)).
Qed.

Global Instance prod_discrete (x : A × B) :
Discrete (x.1) Discrete (x.2) Discrete x.
Proof. by intros ???[??]; split; apply (discrete_0 _). Qed.
Global Instance prod_ofe_discrete :
OfeDiscrete A OfeDiscrete B OfeDiscrete prodO.
Proof. intros ?? [??]; apply _. Qed.

Lemma pair_dist n (a1 a2 : A) (b1 b2 : B) :
(a1, b1) ≡{n}≡ (a2, b2) a1 ≡{n}≡ a2 b1 ≡{n}≡ b2.
Proof. reflexivity. Qed.
End product.

Global Arguments prodO : clear implicits.

Below we make prod_dist type class opaque, so we first lift all instances
Global Instance pair_ne {A B : ofe} : NonExpansive2 (@pair A B) := _.
Global Instance pair_dist_inj {A B : ofe} n :
Inj2 (≡{n}≡) (≡{n}≡) (≡{n}≡) (@pair A B) := _.
Global Instance fst_ne {A B : ofe} : NonExpansive (@fst A B) := _.
Global Instance snd_ne {A B : ofe} : NonExpansive (@snd A B) := _.

Global Instance curry_ne {A B C : ofe} n :
Proper (((≡{n}@{A×B}≡) ==> (≡{n}@{C}≡)) ==>
(≡{n}≡) ==> (≡{n}≡) ==> (≡{n}≡)) curry := _.
Global Instance uncurry_ne {A B C : ofe} n :
Proper (((≡{n}≡) ==> (≡{n}≡) ==> (≡{n}≡)) ==>
(≡{n}@{A×B}≡) ==> (≡{n}@{C}≡)) uncurry := _.

Global Instance curry3_ne {A B C D : ofe} n :
Proper (((≡{n}@{A×B×C}≡) ==> (≡{n}@{D}≡)) ==>
(≡{n}≡) ==> (≡{n}≡) ==> (≡{n}≡) ==> (≡{n}≡)) curry3 := _.
Global Instance uncurry3_ne {A B C D : ofe} n :
Proper (((≡{n}≡) ==> (≡{n}≡) ==> (≡{n}≡) ==> (≡{n}≡)) ==>
(≡{n}@{A×B×C}≡) ==> (≡{n}@{D}≡)) uncurry3 := _.

Global Instance curry4_ne {A B C D E : ofe} n :
Proper (((≡{n}@{A×B×C×D}≡) ==> (≡{n}@{E}≡)) ==>
(≡{n}≡) ==> (≡{n}≡) ==> (≡{n}≡) ==> (≡{n}≡) ==> (≡{n}≡)) curry4 := _.
Global Instance uncurry4_ne {A B C D E : ofe} n :
Proper (((≡{n}≡) ==> (≡{n}≡) ==> (≡{n}≡) ==> (≡{n}≡) ==> (≡{n}≡)) ==>
(≡{n}@{A×B×C×D}≡) ==> (≡{n}@{E}≡)) uncurry4 := _.

Global Typeclasses Opaque prod_dist.

Global Instance prod_map_ne {A A' B B' : ofe} n :
Proper ((dist n ==> dist n) ==> (dist n ==> dist n) ==>
dist n ==> dist n) (@prod_map A A' B B').
Proof. by intros f f' Hf g g' Hg ?? [??]; split; [apply Hf|apply Hg]. Qed.
Definition prodO_map {A A' B B'} (f : A -n> A') (g : B -n> B') :
prodO A B -n> prodO A' B' := OfeMor (prod_map f g).
Global Instance prodO_map_ne {A A' B B'} :
NonExpansive2 (@prodO_map A A' B B').
Proof. intros n f f' Hf g g' Hg [??]; split; [apply Hf|apply Hg]. Qed.

# COFE → OFE Functors

Record oFunctor := OFunctor {
oFunctor_car : A `{!Cofe A} B `{!Cofe B}, ofe;
oFunctor_map `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} :
((A2 -n> A1) × (B1 -n> B2)) oFunctor_car A1 B1 -n> oFunctor_car A2 B2;
oFunctor_map_ne `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} :
NonExpansive (@oFunctor_map A1 _ A2 _ B1 _ B2 _);
oFunctor_map_id `{!Cofe A, !Cofe B} (x : oFunctor_car A B) :
oFunctor_map (cid,cid) x x;
oFunctor_map_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 :
oFunctor_map (fg, g'f') x oFunctor_map (g,g') (oFunctor_map (f,f') x)
}.
Global Existing Instance oFunctor_map_ne.
Global Instance: Params (@oFunctor_map) 9 := {}.

Declare Scope oFunctor_scope.
Delimit Scope oFunctor_scope with OF.
Bind Scope oFunctor_scope with oFunctor.

Class oFunctorContractive (F : oFunctor) :=
#[global] oFunctor_map_contractive `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} ::
Contractive (@oFunctor_map F A1 _ A2 _ B1 _ B2 _).
Global Hint Mode oFunctorContractive ! : typeclass_instances.

Not a coercion due to the Cofe type class argument, and to avoid ambiguous coercion paths, see https://gitlab.mpi-sws.org/iris/iris/issues/240.
Definition oFunctor_apply (F: oFunctor) (A: ofe) `{!Cofe A} : ofe :=
oFunctor_car F A A.

Program Definition oFunctor_oFunctor_compose (F1 F2 : oFunctor)
`{!∀ `{Cofe A, Cofe B}, Cofe (oFunctor_car F2 A B)} : oFunctor := {|
oFunctor_car A _ B _ := oFunctor_car F1 (oFunctor_car F2 B A) (oFunctor_car F2 A B);
oFunctor_map A1 _ A2 _ B1 _ B2 _ 'fg :=
oFunctor_map F1 (oFunctor_map F2 (fg.2,fg.1),oFunctor_map F2 fg)
|}.
Next Obligation.
intros F1 F2 ? A1 ? A2 ? B1 ? B2 ? n [f1 g1] [f2 g2] [??]; simpl in ×.
apply oFunctor_map_ne; split; apply oFunctor_map_ne; by split.
Qed.
Next Obligation.
intros F1 F2 ? A ? B ? x; simpl in ×. rewrite -{2}(oFunctor_map_id F1 x).
apply equiv_distn. apply oFunctor_map_ne.
splity /=; by rewrite !oFunctor_map_id.
Qed.
Next Obligation.
intros F1 F2 ? A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x; simpl in ×.
rewrite -oFunctor_map_compose. apply equiv_distn. apply oFunctor_map_ne.
splity /=; by rewrite !oFunctor_map_compose.
Qed.
Global Instance oFunctor_oFunctor_compose_contractive_1 (F1 F2 : oFunctor)
`{!∀ `{Cofe A, Cofe B}, Cofe (oFunctor_car F2 A B)} :
oFunctorContractive F1 oFunctorContractive (oFunctor_oFunctor_compose F1 F2).
Proof.
intros ? A1 ? A2 ? B1 ? B2 ? n [f1 g1] [f2 g2] Hfg; simpl in ×.
f_contractive; destruct Hfg; split; simpl in *; apply oFunctor_map_ne; by split.
Qed.
Global Instance oFunctor_oFunctor_compose_contractive_2 (F1 F2 : oFunctor)
`{!∀ `{Cofe A, Cofe B}, Cofe (oFunctor_car F2 A B)} :
oFunctorContractive F2 oFunctorContractive (oFunctor_oFunctor_compose F1 F2).
Proof.
intros ? A1 ? A2 ? B1 ? B2 ? n [f1 g1] [f2 g2] Hfg; simpl in ×.
f_equiv; split; simpl in *; f_contractive; destruct Hfg; by split.
Qed.

Program Definition constOF (B : ofe) : oFunctor :=
{| oFunctor_car A1 A2 _ _ := B; oFunctor_map A1 _ A2 _ B1 _ B2 _ f := cid |}.
Solve Obligations with done.
Coercion constOF : ofe >-> oFunctor.

Global Instance constOF_contractive B : oFunctorContractive (constOF B).
Proof. rewrite /oFunctorContractive; apply _. Qed.

Program Definition idOF : oFunctor :=
{| oFunctor_car A1 _ A2 _ := A2; oFunctor_map A1 _ A2 _ B1 _ B2 _ f := f.2 |}.
Solve Obligations with done.
Notation "∙" := idOF : oFunctor_scope.

Program Definition prodOF (F1 F2 : oFunctor) : oFunctor := {|
oFunctor_car A _ B _ := prodO (oFunctor_car F1 A B) (oFunctor_car F2 A B);
oFunctor_map A1 _ A2 _ B1 _ B2 _ fg :=
prodO_map (oFunctor_map F1 fg) (oFunctor_map F2 fg)
|}.
Next Obligation.
intros ?? A1 ? A2 ? B1 ? B2 ? n ???; by apply prodO_map_ne; apply oFunctor_map_ne.
Qed.
Next Obligation. by intros F1 F2 A ? B ? [??]; rewrite /= !oFunctor_map_id. Qed.
Next Obligation.
intros F1 F2 A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' [??]; simpl.
by rewrite !oFunctor_map_compose.
Qed.
Notation "F1 * F2" := (prodOF F1%OF F2%OF) : oFunctor_scope.

Global Instance prodOF_contractive F1 F2 :
oFunctorContractive F1 oFunctorContractive F2
oFunctorContractive (prodOF F1 F2).
Proof.
intros ?? A1 ? A2 ? B1 ? B2 ? n ???;
by apply prodO_map_ne; apply oFunctor_map_contractive.
Qed.

Program Definition ofe_morOF (F1 F2 : oFunctor) : oFunctor := {|
oFunctor_car A _ B _ := oFunctor_car F1 B A -n> oFunctor_car F2 A B;
oFunctor_map A1 _ A2 _ B1 _ B2 _ fg :=
ofe_morO_map (oFunctor_map F1 (fg.2, fg.1)) (oFunctor_map F2 fg)
|}.
Next Obligation.
intros F1 F2 A1 ? A2 ? B1 ? B2 ? n [f g] [f' g'] Hfg; simpl in ×.
apply ofe_morO_map_ne; apply oFunctor_map_ne; split; by apply Hfg.
Qed.
Next Obligation.
intros F1 F2 A ? B ? [f ?] ?; simpl. rewrite /= !oFunctor_map_id.
apply (ne_proper f). apply oFunctor_map_id.
Qed.
Next Obligation.
intros F1 F2 A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' [h ?] ?; simpl in ×.
rewrite -!oFunctor_map_compose. do 2 apply (ne_proper _). apply oFunctor_map_compose.
Qed.
Notation "F1 -n> F2" := (ofe_morOF F1%OF F2%OF) : oFunctor_scope.

Global Instance ofe_morOF_contractive F1 F2 :
oFunctorContractive F1 oFunctorContractive F2
oFunctorContractive (ofe_morOF F1 F2).
Proof.
intros ?? A1 ? A2 ? B1 ? B2 ? n [f g] [f' g'] Hfg; simpl in ×.
apply ofe_morO_map_ne; apply oFunctor_map_contractive;
split; intros m Hlt; split; simpl.
all: destruct Hfg as [Hfg]; destruct (Hfg m); auto.
Qed.

# Sum type

Section sum.
Context {A B : ofe}.

Local Instance sum_dist : Dist (A + B) := λ n, sum_relation (dist n) (dist n).
Global Instance inl_ne : NonExpansive (@inl A B) := _.
Global Instance inr_ne : NonExpansive (@inr A B) := _.
Global Instance inl_ne_inj n : Inj (dist n) (dist n) (@inl A B) := _.
Global Instance inr_ne_inj n : Inj (dist n) (dist n) (@inr A B) := _.

Definition sum_ofe_mixin : OfeMixin (A + B).
Proof.
split.
- intros x y; splitHx.
+ destruct Hxn; constructor; by apply equiv_dist.
+ destruct (Hx 0); constructor; apply equiv_distn; by apply (inj _).
- apply _.
- destruct 1; constructor; eapply dist_lt; eauto.
Qed.
Canonical Structure sumO : ofe := Ofe (A + B) sum_ofe_mixin.

Program Definition inl_chain (c : chain sumO) (a : A) : chain A :=
{| chain_car n := match c n return _ with inl a'a' | _a end |}.
Next Obligation. intros c a n i ?; simpl. by destruct (chain_cauchy c n i). Qed.
Program Definition inr_chain (c : chain sumO) (b : B) : chain B :=
{| chain_car n := match c n return _ with inr b'b' | _b end |}.
Next Obligation. intros c b n i ?; simpl. by destruct (chain_cauchy c n i). Qed.

Definition sum_compl `{!Cofe A, !Cofe B} : Compl sumO := λ c,
match c 0 with
| inl ainl (compl (inl_chain c a))
| inr binr (compl (inr_chain c b))
end.
Global Program Instance sum_cofe `{Cofe A, Cofe B} : Cofe sumO :=
{ compl := sum_compl }.
Next Obligation.
intros ?? n c; rewrite /compl /sum_compl.
oinversion (chain_cauchy c 0 n); first by si_solver.
- rewrite (conv_compl n (inl_chain c _)) /=. destruct (c n); naive_solver.
- rewrite (conv_compl n (inr_chain c _)) /=. destruct (c n); naive_solver.
Qed.

Global Instance inl_discrete (x : A) : Discrete x Discrete (inl x).
Proof. inversion_clear 2; constructor; by apply (discrete_0 _). Qed.
Global Instance inr_discrete (y : B) : Discrete y Discrete (inr y).
Proof. inversion_clear 2; constructor; by apply (discrete_0 _). Qed.
Global Instance sum_ofe_discrete :
OfeDiscrete A OfeDiscrete B OfeDiscrete sumO.
Proof. intros ?? [?|?]; apply _. Qed.
End sum.

Global Arguments sumO : clear implicits.
Global Typeclasses Opaque sum_dist.

Global Instance sum_map_ne {A A' B B' : ofe} n :
Proper ((dist n ==> dist n) ==> (dist n ==> dist n) ==>
dist n ==> dist n) (@sum_map A A' B B').
Proof.
intros f f' Hf g g' Hg ??; destruct 1; constructor; [by apply Hf|by apply Hg].
Qed.
Definition sumO_map {A A' B B'} (f : A -n> A') (g : B -n> B') :
sumO A B -n> sumO A' B' := OfeMor (sum_map f g).
Global Instance sumO_map_ne {A A' B B'} :
NonExpansive2 (@sumO_map A A' B B').
Proof. intros n f f' Hf g g' Hg [?|?]; constructor; [apply Hf|apply Hg]. Qed.

Program Definition sumOF (F1 F2 : oFunctor) : oFunctor := {|
oFunctor_car A _ B _ := sumO (oFunctor_car F1 A B) (oFunctor_car F2 A B);
oFunctor_map A1 _ A2 _ B1 _ B2 _ fg :=
sumO_map (oFunctor_map F1 fg) (oFunctor_map F2 fg)
|}.
Next Obligation.
intros ?? A1 ? A2 ? B1 ? B2 ? n ???; by apply sumO_map_ne; apply oFunctor_map_ne.
Qed.
Next Obligation. by intros F1 F2 A ? B ? [?|?]; rewrite /= !oFunctor_map_id. Qed.
Next Obligation.
intros F1 F2 A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' [?|?]; simpl;
by rewrite !oFunctor_map_compose.
Qed.
Notation "F1 + F2" := (sumOF F1%OF F2%OF) : oFunctor_scope.

Global Instance sumOF_contractive F1 F2 :
oFunctorContractive F1 oFunctorContractive F2
oFunctorContractive (sumOF F1 F2).
Proof.
intros ?? A1 ? A2 ? B1 ? B2 ? n ???;
by apply sumO_map_ne; apply oFunctor_map_contractive.
Qed.

# Discrete OFEs

Section discrete_ofe.
Context {A : Type} `{!Equiv A} (Heq : @Equivalence A (≡)).

Local Instance discrete_dist : Dist A := λ n x y, x y.
Definition discrete_ofe_mixin : OfeMixin A.
Proof using Type×.
split.
- intros x y; split; [done|intros Hn; apply (Hn 0)].
- done.
- done.
Qed.

Global Instance discrete_ofe_discrete : OfeDiscrete (Ofe A discrete_ofe_mixin).
Proof. by intros x y. Qed.

Global Program Instance discrete_cofe : Cofe (Ofe A discrete_ofe_mixin) :=
{ compl c := c 0 }.
Next Obligation.
intros n c. rewrite /compl /=;
symmetry; apply (chain_cauchy c 0 n). lia.
Qed.
End discrete_ofe.

The combinators discreteO and leibnizO should be used with care. There are two ways in which they can be used:
1. To define an OFE on a ground type, such as nat, expr, etc. The OFE instance should be defined as Canonical Structure tyO := leibnizO ty or Canonical Structure tyO := discreteO ty, so not using Definition. See natO below for an example. Make sure to avoid overlapping instances, so always check if no instance has already been defined. For most of the types from Coq, std++, and Iris, instances are present in Iris. The convention is to use the name tyO for the OFE instance of a type ty. 2. As part of abstractions that are parametrized with a Type, but where an ofe is needed to use (camera) combinators. See ghost_var as an example. In this case, the public API of the abstraction should exclusively use Type, i.e., the use of leibnizO or discreteO should not leak. Otherwise client code can end up with overlapping instances, and thus experience odd unification failures.
You should *never* use leibnizO or discreteO on compound types such as list nat. That creates overlapping canonical instances for the head symbol (e.g., listO and leibnizO (list nat)) and confuses unification. Instead, you have two options:
• declare/use a canonical instance for the ground type, e.g., listO natO.
• declare a newtype, e.g., Record ty := Ty { ty_car : list nat }, and then declare a canonical instance for that type, e.g., Canonical Structure tyO := leibnizO ty.
The combinator discreteO A lifts an existing Equiv A instance into a discrete OFE.
Notation discreteO A := (Ofe A (discrete_ofe_mixin _)).

The combinator leibnizO A lifts Leibniz equality = into a discrete OFE. The implementation forces the Equivalence proof to be eq_equivalence so that Coq does not accidentally use another one, like ofe_equivalence, in the case of aliases. See also https://gitlab.mpi-sws.org/iris/iris/issues/299
In order to define a discrete CMRA with carrier A (in the file cmra.v) we need to determine the Equivalence A proof that was used to construct the OFE instance of A (note that this proof is not the same as the one we obtain via ofe_equivalence).
We obtain the proof of Equivalence A by inferring the canonical OFE mixin using ofe_mixin_of A, and then check whether it is indeed a discrete OFE. This will fail if no OFE, or an OFE other than the discrete OFE, was registered.
Notation discrete_ofe_equivalence_of A := ltac:(
match constr:(ofe_mixin_of A) with
| discrete_ofe_mixin ?Hexact H
end) (only parsing).

Global Instance leibnizO_leibniz A : LeibnizEquiv (leibnizO A).
Proof. by intros x y. Qed.

# Basic Coq types

Canonical Structure boolO := leibnizO bool.
Canonical Structure natO := leibnizO nat.
Canonical Structure positiveO := leibnizO positive.
Canonical Structure NO := leibnizO N.
Canonical Structure ZO := leibnizO Z.

Section prop.
Local Instance Prop_equiv : Equiv Prop := iff.
Local Instance Prop_equivalence : Equivalence (≡@{Prop}) := _.
Canonical Structure PropO := discreteO Prop.
End prop.

# Option type

Section option.
Context {A : ofe}.

Local Instance option_dist : Dist (option A) := λ n, option_Forall2 (dist n).
Lemma option_dist_Forall2 n mx my : mx ≡{n}≡ my option_Forall2 (dist n) mx my.
Proof. done. Qed.

Definition option_ofe_mixin : OfeMixin (option A).
Proof.
split.
- intros mx my; split; [by destruct 1; constructor; apply equiv_dist|].
intros Hxy; destruct (Hxy 0); constructor; apply equiv_dist.
by intros n; oinversion (Hxy n).
- apply _.
- destruct 1; constructor; eauto using dist_le with si_solver.
Qed.
Canonical Structure optionO := Ofe (option A) option_ofe_mixin.

Program Definition option_chain (c : chain optionO) (x : A) : chain A :=
{| chain_car n := default x (c n) |}.
Next Obligation. intros c x n i ?; simpl. by destruct (chain_cauchy c n i). Qed.
Definition option_compl `{!Cofe A} : Compl optionO := λ c,
match c 0 with Some xSome (compl (option_chain c x)) | NoneNone end.
Global Program Instance option_cofe `{Cofe A} : Cofe optionO :=
{ compl := option_compl }.
Next Obligation.
intros ? n c; rewrite /compl /option_compl.
oinversion (chain_cauchy c 0 n); auto with lia; [].
constructor. rewrite (conv_compl n (option_chain c _)) /=.
destruct (c n); naive_solver.
Qed.

Global Instance option_ofe_discrete : OfeDiscrete A OfeDiscrete optionO.
Proof. destruct 2; constructor; by apply (discrete_0 _). Qed.

Global Instance Some_ne : NonExpansive (@Some A).
Proof. by constructor. Qed.
Global Instance is_Some_ne n : Proper (dist n ==> iff) (@is_Some A).
Proof. destruct 1; split; eauto. Qed.
Global Instance Some_dist_inj n : Inj (dist n) (dist n) (@Some A).
Proof. by inversion_clear 1. Qed.
Global Instance from_option_ne {B} (R : relation B) n :
Proper ((dist (A:=A) n ==> R) ==> R ==> dist n ==> R) from_option.
Proof. destruct 3; simpl; auto. Qed.

Global Instance None_discrete : Discrete (@None A).
Proof. inversion_clear 1; constructor. Qed.
Global Instance Some_discrete x : Discrete x Discrete (Some x).
Proof. by intros ?; inversion_clear 1; constructor; apply discrete_0. Qed.

Lemma dist_None n mx : mx ≡{n}≡ None mx = None.
Proof. split; [by inversion_clear 1|by intros ->]. Qed.
Lemma dist_Some_inv_l n mx my x :
mx ≡{n}≡ my mx = Some x y, my = Some y x ≡{n}≡ y.
Proof. destruct 1; naive_solver. Qed.
Lemma dist_Some_inv_r n mx my y :
mx ≡{n}≡ my my = Some y x, mx = Some x x ≡{n}≡ y.
Proof. destruct 1; naive_solver. Qed.
Lemma dist_Some_inv_l' n my x : Some x ≡{n}≡ my x', Some x' = my x ≡{n}≡ x'.
Proof. intros ?%(dist_Some_inv_l _ _ _ x); naive_solver. Qed.
Lemma dist_Some_inv_r' n mx y : mx ≡{n}≡ Some y y', mx = Some y' y ≡{n}≡ y'.
Proof. intros ?%(dist_Some_inv_r _ _ _ y); naive_solver. Qed.
End option.

Global Typeclasses Opaque option_dist.
Global Arguments optionO : clear implicits.

Global Instance option_fmap_ne {A B : ofe} n:
Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@fmap option _ A B).
Proof. intros f f' Hf ?? []; constructor; auto. Qed.
Global Instance option_mbind_ne {A B : ofe} n:
Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@mbind option _ A B).
Proof. destruct 2; simpl; auto. Qed.
Global Instance option_mjoin_ne {A : ofe} n:
Proper (dist n ==> dist n) (@mjoin option _ A).
Proof. destruct 1 as [?? []|]; simpl; by constructor. Qed.

Global Instance option_fmap_dist_inj {A B : ofe} (f : A B) n :
Inj (≡{n}≡) (≡{n}≡) f Inj (≡{n}@{option A}≡) (≡{n}@{option B}≡) (fmap f).
Proof. apply option_fmap_inj. Qed.

Lemma fmap_Some_dist {A B : ofe} (f : A B) (mx : option A) (y : B) n :
f <\$> mx ≡{n}≡ Some y x : A, mx = Some x y ≡{n}≡ f x.
Proof.
split; [|by intros (x&->&->)].
intros (?&?%fmap_Some&?)%dist_Some_inv_r'; naive_solver.
Qed.

Definition optionO_map {A B} (f : A -n> B) : optionO A -n> optionO B :=
OfeMor (fmap f : optionO A optionO B).
Global Instance optionO_map_ne A B : NonExpansive (@optionO_map A B).
Proof. by intros n f f' Hf []; constructor; apply Hf. Qed.

Program Definition optionOF (F : oFunctor) : oFunctor := {|
oFunctor_car A _ B _ := optionO (oFunctor_car F A B);
oFunctor_map A1 _ A2 _ B1 _ B2 _ fg := optionO_map (oFunctor_map F fg)
|}.
Next Obligation.
by intros F A1 ? A2 ? B1 ? B2 ? n f g Hfg; apply optionO_map_ne, oFunctor_map_ne.
Qed.
Next Obligation.
intros F A ? B ? x. rewrite /= -{2}(option_fmap_id x).
apply option_fmap_equiv_exty; apply oFunctor_map_id.
Qed.
Next Obligation.
intros F A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x. rewrite /= -option_fmap_compose.
apply option_fmap_equiv_exty; apply oFunctor_map_compose.
Qed.

Global Instance optionOF_contractive F :
oFunctorContractive F oFunctorContractive (optionOF F).
Proof.
by intros ? A1 ? A2 ? B1 ? B2 ? n f g Hfg;
apply optionO_map_ne, oFunctor_map_contractive.
Qed.

# Later type

Note that the projection later_car is not non-expansive (see also the lemma later_car_anti_contractive below), so it cannot be used in the logic. If you need to get a witness out, you should use the lemma Next_uninj instead.
Record later (A : Type) : Type := Next { later_car : A }.
Add Printing Constructor later.
Global Arguments Next {_} _.
Global Arguments later_car {_} _.
Global Instance: Params (@Next) 1 := {}.

Section later.
Context {A : ofe}.
Local Instance later_equiv : Equiv (later A) := λ x y, later_car x later_car y.
Local Instance later_dist : Dist (later A) := λ n x y,
dist_later n (later_car x) (later_car y).
Definition later_ofe_mixin : OfeMixin (later A).
Proof.
split.
- intros x y; unfold equiv, later_equiv; rewrite !equiv_dist.
split; intros Hxy n; [done|].
eapply (Hxy (S n)). lia.
- split; rewrite /dist /later_dist.
+ by intros [x].
+ by intros [x] [y].
+ by intros [x] [y] [z] ??; trans y.
- intros n m [x] [y] Hdist ?; split; intros p Hp. eapply Hdist; by trans m.
Qed.
Canonical Structure laterO : ofe := Ofe (later A) later_ofe_mixin.

Program Definition later_chain (c : chain laterO) : chain A :=
{| chain_car n := later_car (c (S n)) |}.
Next Obligation. intros c n i ?; apply (chain_cauchy c (S n)); lia. Qed.
Global Program Instance later_cofe `{Cofe A} : Cofe laterO :=
{ compl c := Next (compl (later_chain c)) }.
Next Obligation.
intros ? n c. apply dist_later_fin_iff.
destruct n as [|n]; [done|by apply (conv_compl n (later_chain c))].
Qed.

Global Instance Next_contractive : Contractive (@Next A).
Proof. by intros n x y. Qed.
Global Instance Next_inj n : Inj (dist_later n) (dist n) (@Next A).
Proof. by intros x y. Qed.

Lemma Next_uninj x : a, x Next a.
Proof. by (later_car x). Qed.
Local Instance later_car_anti_contractive n :
Proper (dist n ==> dist_later n) later_car.
Proof. move⇒ [x] [y] /= Hxy. done. Qed.

f is contractive iff it can factor into Next and a non-expansive function.
Lemma contractive_alt {B : ofe} (f : A B) :
Contractive f g : later A B, NonExpansive g x, f x g (Next x).
Proof.
split.
- intros Hf. (f later_car); split⇒ // n x y ?. by f_equiv.
- intros (g&Hg&Hf) n x y Hxy. rewrite !Hf. by apply Hg.
Qed.
End later.

Global Arguments laterO : clear implicits.

Definition later_map {A B} (f : A B) (x : later A) : later B :=
Next (f (later_car x)).
Global Instance later_map_ne {A B : ofe} (f : A B) n :
Proper (dist_later n ==> dist_later n) f
Proper (dist n ==> dist n) (later_map f) | 0.
Proof.
intros P [x] [y] Hdist; rewrite /later_map //=.
split; intros m Hm; apply P, Hm. apply Hdist.
Qed.
Global Instance later_map_ne' {A B : ofe} (f : A B) `{NonExpansive f} :
NonExpansive (later_map f).
Proof.
intros ? [x] [y] Hdist. unfold later_map; simpl.
split; intros ??; simpl. f_equiv. by eapply Hdist.
Qed.
Global Instance later_map_proper {A B : ofe} (f : A B) :
Proper ((≡) ==> (≡)) f
Proper ((≡) ==> (≡)) (later_map f).
Proof. solve_proper. Qed.

Lemma later_map_Next {A B : ofe} (f : A B) x :
later_map f (Next x) = Next (f x).
Proof. done. Qed.
Lemma later_map_id {A} (x : later A) : later_map id x = x.
Proof. by destruct x. Qed.
Lemma later_map_compose {A B C} (f : A B) (g : B C) (x : later A) :
later_map (g f) x = later_map g (later_map f x).
Proof. by destruct x. Qed.
Lemma later_map_ext {A B : ofe} (f g : A B) x :
( x, f x g x) later_map f x later_map g x.
Proof. destruct x; intros Hf; apply Hf. Qed.
Definition laterO_map {A B} (f : A -n> B) : laterO A -n> laterO B :=
OfeMor (later_map f).
Global Instance laterO_map_contractive (A B : ofe) : Contractive (@laterO_map A B).
Proof. intros n f g Hlater [x]; split; intros ??; simpl. by apply Hlater. Qed.

Program Definition laterOF (F : oFunctor) : oFunctor := {|
oFunctor_car A _ B _ := laterO (oFunctor_car F A B);
oFunctor_map A1 _ A2 _ B1 _ B2 _ fg := laterO_map (oFunctor_map F fg)
|}.
Next Obligation.
intros F A1 ? A2 ? B1 ? B2 ? n fg fg' ?.
by apply (contractive_ne laterO_map), oFunctor_map_ne.
Qed.
Next Obligation.
intros F A ? B ? x; simpl. rewrite -{2}(later_map_id x).
apply later_map_exty. by rewrite oFunctor_map_id.
Qed.
Next Obligation.
intros F A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x; simpl. rewrite -later_map_compose.
apply later_map_exty; apply oFunctor_map_compose.
Qed.
Notation "▶ F" := (laterOF F%OF) (at level 20, right associativity) : oFunctor_scope.

Global Instance laterOF_contractive F : oFunctorContractive (laterOF F).
Proof.
intros A1 ? A2 ? B1 ? B2 ? n fg fg' Hfg. apply laterO_map_contractive.
split; intros ???; simpl. by eapply oFunctor_map_ne, Hfg.
Qed.

# Dependently-typed functions over a discrete domain

This separate notion is useful whenever we need dependent functions, and whenever we want to avoid the hassle of the bundled non-expansive function type.
Note that non-dependent functions over a discrete domain, A -d> B (following the notation we introduce below) are non-expansive if they are Proper ((≡) ==> (≡)). In other words, since the domain is discrete, non-expansiveness and respecting (≡) are the same. If the domain is moreover Leibniz (LeibnizEquiv A), we get both for free.
We make discrete_fun a definition so that we can register it as a canonical structure. We do not bundle the Proper proof to keep discrete_fun easier to use. It turns out all the desired OFE and functorial properties do not rely on this Proper instance.
Definition discrete_fun {A} (B : A ofe) := x : A, B x.

Section discrete_fun.
Context {A : Type} {B : A ofe}.
Implicit Types f g : discrete_fun B.

Local Instance discrete_fun_equiv : Equiv (discrete_fun B) := λ f g, x, f x g x.
Local Instance discrete_fun_dist : Dist (discrete_fun B) := λ n f g, x, f x ≡{n}≡ g x.
Definition discrete_fun_ofe_mixin : OfeMixin (discrete_fun B).
Proof.
split.
- intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
intros Hfg k; apply equiv_distn; apply Hfg.
- intros n; split.
+ by intros f x.
+ by intros f g ? x.
+ by intros f g h ?? x; trans (g x).
- by intros n m f g ? ? x; eauto using dist_le with si_solver.
Qed.
Canonical Structure discrete_funO := Ofe (discrete_fun B) discrete_fun_ofe_mixin.

Program Definition discrete_fun_chain `(c : chain discrete_funO)
(x : A) : chain (B x) := {| chain_car n := c n x |}.
Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed.
Global Program Instance discrete_fun_cofe `{ x, Cofe (B x)} : Cofe discrete_funO :=
{ compl c x := compl (discrete_fun_chain c x) }.
Next Obligation. intros ? n c x. apply (conv_compl n (discrete_fun_chain c x)). Qed.

Global Instance discrete_fun_inhabited `{ x, Inhabited (B x)} : Inhabited discrete_funO :=
populate (λ _, inhabitant).
Global Instance discrete_fun_lookup_discrete `{EqDecision A} f x :
Discrete f Discrete (f x).
Proof.
intros Hf y ?.
set (g x' := if decide (x = x') is left H then eq_rect _ B y _ H else f x').
trans (g x).
{ apply Hfx'. unfold g. by destruct (decide _) as [[]|]. }
unfold g. destruct (decide _) as [Hx|]; last done.
by rewrite (proof_irrel Hx eq_refl).
Qed.
End discrete_fun.

Global Arguments discrete_funO {_} _.
Notation "A -d> B" :=
(@discrete_funO A (λ _, B)) (at level 99, B at level 200, right associativity).

Definition discrete_fun_map {A} {B1 B2 : A ofe} (f : x, B1 x B2 x)
(g : discrete_fun B1) : discrete_fun B2 := λ x, f _ (g x).

Lemma discrete_fun_map_ext {A} {B1 B2 : A ofe} (f1 f2 : x, B1 x B2 x)
(g : discrete_fun B1) :
( x, f1 x (g x) f2 x (g x)) discrete_fun_map f1 g discrete_fun_map f2 g.
Proof. done. Qed.
Lemma discrete_fun_map_id {A} {B : A ofe} (g : discrete_fun B) :
discrete_fun_map (λ _, id) g = g.
Proof. done. Qed.
Lemma discrete_fun_map_compose {A} {B1 B2 B3 : A ofe}
(f1 : x, B1 x B2 x) (f2 : x, B2 x B3 x) (g : discrete_fun B1) :
discrete_fun_map (λ x, f2 x f1 x) g = discrete_fun_map f2 (discrete_fun_map f1 g).
Proof. done. Qed.

Global Instance discrete_fun_map_ne {A} {B1 B2 : A ofe} (f : x, B1 x B2 x) n :
( x, Proper (dist n ==> dist n) (f x))
Proper (dist n ==> dist n) (discrete_fun_map f).
Proof. by intros ? y1 y2 Hy x; rewrite /discrete_fun_map (Hy x). Qed.

Definition discrete_funO_map {A} {B1 B2 : A ofe} (f : discrete_fun (λ x, B1 x -n> B2 x)) :
discrete_funO B1 -n> discrete_funO B2 := OfeMor (discrete_fun_map f).
Global Instance discrete_funO_map_ne {A} {B1 B2 : A ofe} :
NonExpansive (@discrete_funO_map A B1 B2).
Proof. intros n f1 f2 Hf g x; apply Hf. Qed.

Program Definition discrete_funOF {C} (F : C oFunctor) : oFunctor := {|
oFunctor_car A _ B _ := discrete_funO (λ c, oFunctor_car (F c) A B);
oFunctor_map A1 _ A2 _ B1 _ B2 _ fg := discrete_funO_map (λ c, oFunctor_map (F c) fg)
|}.
Next Obligation.
intros C F A1 ? A2 ? B1 ? B2 ? n ?? g.
by apply discrete_funO_map_ne=>?; apply oFunctor_map_ne.
Qed.
Next Obligation.
intros C F A ? B ? g; simpl. rewrite -{2}(discrete_fun_map_id g).
apply discrete_fun_map_exty; apply oFunctor_map_id.
Qed.
Next Obligation.
intros C F A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f1 f2 f1' f2' g.
rewrite /= -discrete_fun_map_compose.
apply discrete_fun_map_exty; apply oFunctor_map_compose.
Qed.

Notation "T -d> F" := (@discrete_funOF T%type (λ _, F%OF)) : oFunctor_scope.

Global Instance discrete_funOF_contractive {C} (F : C oFunctor) :
( c, oFunctorContractive (F c)) oFunctorContractive (discrete_funOF F).
Proof.
intros ? A1 ? A2 ? B1 ? B2 ? n ?? g.
by apply discrete_funO_map_nec; apply oFunctor_map_contractive.
Qed.

# Constructing isomorphic OFEs

Lemma iso_ofe_mixin {A : ofe} {B : Type} `{!Equiv B, !Dist B} (g : B A)
(g_equiv : y1 y2, y1 y2 g y1 g y2)
(g_dist : n y1 y2, y1 ≡{n}≡ y2 g y1 ≡{n}≡ g y2) : OfeMixin B.
Proof.
split.
- intros y1 y2. rewrite g_equiv. setoid_rewrite g_dist. apply equiv_dist.
- split.
+ intros y. by apply g_dist.
+ intros y1 y2. by rewrite !g_dist.
+ intros y1 y2 y3. rewrite !g_dist. intros ??; etrans; eauto.
- intros n m y1 y2. rewrite !g_dist. eauto using dist_le with si_solver.
Qed.

Section iso_cofe_subtype.
Context {A B : ofe} `{Cofe A} (P : A Prop) (f : x, P x B) (g : B A).
Context (g_dist : n y1 y2, y1 ≡{n}≡ y2 g y1 ≡{n}≡ g y2).
Let Hgne : NonExpansive g.
Proof. intros n y1 y2. apply g_dist. Defined.
Local Existing Instance Hgne.
Context (gf : x Hx, g (f x Hx) x).
Context (Hlimit : c : chain B, P (compl (chain_map g c))).
Program Definition iso_cofe_subtype : Cofe B :=
{| compl c := f (compl (chain_map g c)) _ |}.
Next Obligation. apply Hlimit. Qed.
Next Obligation.
intros n c; simpl. apply g_dist. by rewrite gf conv_compl.
Qed.
End iso_cofe_subtype.

Lemma iso_cofe_subtype' {A B : ofe} `{Cofe A}
(P : A Prop) (f : x, P x B) (g : B A)
(Pg : y, P (g y))
(g_dist : n y1 y2, y1 ≡{n}≡ y2 g y1 ≡{n}≡ g y2)
(gf : x Hx, g (f x Hx) x)
(Hlimit : LimitPreserving P) : Cofe B.
Proof. apply: (iso_cofe_subtype P f g)=> // c. apply Hlimit⇒ ?; apply Pg. Qed.

Definition iso_cofe {A B : ofe} `{Cofe A} (f : A B) (g : B A)
(g_dist : n y1 y2, y1 ≡{n}≡ y2 g y1 ≡{n}≡ g y2)
(gf : x, g (f x) x) : Cofe B.
Proof. by apply (iso_cofe_subtype (λ _, True) (λ x _, f x) g). Qed.

# Sigma type

Section sigma.
Context {A : ofe} {P : A Prop}.
Implicit Types x : sig P.

Local Instance sig_equiv : Equiv (sig P) := λ x1 x2, `x1 `x2.
Local Instance sig_dist : Dist (sig P) := λ n x1 x2, `x1 ≡{n}≡ `x2.

Definition sig_equiv_def x y : (x y) = (`x `y) := reflexivity _.
Definition sig_dist_def n x y : (x ≡{n}≡ y) = (`x ≡{n}≡ `y) := reflexivity _.

Lemma exist_ne n a1 a2 (H1 : P a1) (H2 : P a2) :
a1 ≡{n}≡ a2 a1 H1 ≡{n}≡ a2 H2.
Proof. done. Qed.

Global Instance proj1_sig_ne : NonExpansive (@proj1_sig _ P).
Proof. by intros n [a Ha] [b Hb] ?. Qed.
Definition sig_ofe_mixin : OfeMixin (sig P).
Proof. by apply (iso_ofe_mixin proj1_sig). Qed.
Canonical Structure sigO : ofe := Ofe (sig P) sig_ofe_mixin.

Global Instance sig_cofe `{!Cofe A, !LimitPreserving P} : Cofe sigO.
Proof. apply (iso_cofe_subtype' P (exist P) proj1_sig)=> //. by intros []. Qed.

Global Instance sig_discrete (x : sig P) : Discrete (`x) Discrete x.
Proof. intros ? y. rewrite sig_dist_def sig_equiv_def. apply (discrete_0 _). Qed.
Global Instance sig_ofe_discrete : OfeDiscrete A OfeDiscrete sigO.
Proof. intros ??. apply _. Qed.
End sigma.

Global Arguments sigO {_} _.

# SigmaT type

Ofe for sigT. The first component must be discrete and use Leibniz equality, while the second component might be any OFE.
Section sigT.
Import EqNotations.

Context {A : Type} {P : A ofe}.
Implicit Types x : sigT P.

The distance for { a : A & P } uses Leibniz equality on A to transport the second components to the same type, and then step-indexed distance on the second component. Unlike in the topos of trees, with (C)OFEs we cannot use step-indexed equality on the first component.
Local Instance sigT_dist : Dist (sigT P) := λ n x1 x2,
Heq : projT1 x1 = projT1 x2, rew Heq in projT2 x1 ≡{n}≡ projT2 x2.

Usually we'd give a direct definition, and show it equivalent to n, x1 ≡{n}≡ x2 when proving the equiv_dist OFE axiom. But here the equivalence requires UIP — see sigT_equiv_eq_alt. By defining equiv in terms of dist, we can define an OFE without assuming UIP, at the cost of complex reasoning on equiv.
Local Instance sigT_equiv : Equiv (sigT P) := λ x1 x2,
n, x1 ≡{n}≡ x2.

Unfolding lemmas. Written with not = to avoid https://github.com/coq/coq/issues/3814.
Definition sigT_equiv_eq x1 x2 : (x1 x2) n, x1 ≡{n}≡ x2 :=
reflexivity _.

Definition sigT_dist_eq x1 x2 n : (x1 ≡{n}≡ x2)
Heq : projT1 x1 = projT1 x2, (rew Heq in projT2 x1) ≡{n}≡ projT2 x2 :=
reflexivity _.

Definition sigT_dist_proj1 n {x y} : x ≡{n}≡ y projT1 x = projT1 y := proj1_ex.
Definition sigT_equiv_proj1 {x y} : x y projT1 x = projT1 y := λ H, proj1_ex (H 0).

Definition sigT_ofe_mixin : OfeMixin (sigT P).
Proof.
split ⇒ // n.
- split; hnf; setoid_rewrite sigT_dist_eq.
+ intros. by eq_refl.
+ move ⇒ [xa x] [ya y] /=. destruct 1 as [-> Heq].
by eq_refl.
+ move ⇒ [xa x] [ya y] [za z] /=.
destruct 1 as [-> Heq1].
destruct 1 as [-> Heq2]. eq_refl ⇒ /=. by trans y.
- setoid_rewrite sigT_dist_eq.
movem [xa x] [ya y] /=. destruct 1 as [-> Heq].
eq_refl. by eapply dist_dist_later.
Qed.

Canonical Structure sigTO : ofe := Ofe (sigT P) sigT_ofe_mixin.

Lemma sigT_equiv_eq_alt `{!∀ a b : A, ProofIrrel (a = b)} x1 x2 :
x1 x2
Heq : projT1 x1 = projT1 x2, rew Heq in projT2 x1 projT2 x2.
Proof.
setoid_rewrite equiv_dist; setoid_rewrite sigT_dist_eq; splitHeq.
- move: (Heq 0) ⇒ [H0eq1 _].
H0eq1n. move: (Heq n) ⇒ [] Hneq1.
by rewrite (proof_irrel H0eq1 Hneq1).
- move: Heq ⇒ [Heq1 Heqn2] n. by Heq1.
Qed.

projT1 is non-expansive and proper.
Global Instance projT1_ne : NonExpansive (projT1 : sigTO leibnizO A).
Proof. solve_proper. Qed.

Global Instance projT1_proper : Proper ((≡) ==> (≡)) (projT1 : sigTO leibnizO A).
Proof. apply ne_proper, projT1_ne. Qed.

projT2 is "non-expansive"; the properness lemma projT2_ne requires UIP.
Lemma projT2_ne n (x1 x2 : sigTO) (Heq : x1 ≡{n}≡ x2) :
rew (sigT_dist_proj1 n Heq) in projT2 x1 ≡{n}≡ projT2 x2.
Proof. by destruct Heq. Qed.

Lemma projT2_proper `{!∀ a b : A, ProofIrrel (a = b)} (x1 x2 : sigTO) (Heqs : x1 x2):
rew (sigT_equiv_proj1 Heqs) in projT2 x1 projT2 x2.
Proof.
move: x1 x2 Heqs ⇒ [a1 x1] [a2 x2] Heqs.
case: (proj1 (sigT_equiv_eq_alt _ _) Heqs) ⇒ /=. intros →.
rewrite (proof_irrel (sigT_equiv_proj1 Heqs) eq_refl) /=. done.
Qed.

existT is "non-expansive" — general, dependently-typed statement.
Lemma existT_ne n {i1 i2} {v1 : P i1} {v2 : P i2} :
(Heq : i1 = i2), (rew f_equal P Heq in v1 ≡{n}≡ v2)
existT i1 v1 ≡{n}≡ existT i2 v2.
Proof. intros ->; simpl. eq_refl ⇒ /=. done. Qed.

Lemma existT_proper {i1 i2} {v1 : P i1} {v2 : P i2} :
(Heq : i1 = i2), (rew f_equal P Heq in v1 v2)
existT i1 v1 existT i2 v2.
Proof. intros Heq Heqv n. apply (existT_ne n Heq), equiv_dist, Heqv. Qed.

existT is "non-expansive" — non-dependently-typed version.
Global Instance existT_ne_2 a : NonExpansive (@existT A P a).
Proof. move ⇒ ??? Heq. apply (existT_ne _ eq_refl Heq). Qed.

Global Instance existT_proper_2 a : Proper ((≡) ==> (≡)) (@existT A P a).
Proof. apply ne_proper, _. Qed.

Implicit Types (c : chain sigTO).

Global Instance sigT_discrete x : Discrete (projT2 x) Discrete x.
Proof.
move: x ⇒ [xa x] ? [ya y] [] /=; intros → ⇒ /= Hxy n.
eq_refl ⇒ /=. apply equiv_dist, (discrete_0 _), Hxy.
Qed.

Global Instance sigT_ofe_discrete : ( a, OfeDiscrete (P a)) OfeDiscrete sigTO.
Proof. intros ??. apply _. Qed.

Lemma sigT_chain_const_proj1 c n : projT1 (c n) = projT1 (c 0).
Proof. refine (sigT_dist_proj1 _ (chain_cauchy c 0 n _)). lia. Qed.

Section cofe.
Context `{!∀ a b : A, ProofIrrel (a = b)} `{!∀ a, Cofe (P a)}.

Program Definition chain_map_snd c : chain (P (projT1 (c 0))) :=
{| chain_car n := rew (sigT_chain_const_proj1 c n) in projT2 (c n) |}.
Next Obligation.
movec n i Hle /=.
case: (chain_cauchy c n i Hle) ⇒ [Heqin Hgoal] /=.
move: (sigT_chain_const_proj1 c i) (sigT_chain_const_proj1 c n)
⇒ Heqi0 Heqn0.
destruct Heqn0.
by rewrite /= (proof_irrel Heqi0 Heqin).
Qed.

Definition sigT_compl : Compl sigTO :=
λ c, existT (projT1 (chain_car c 0)) (compl (chain_map_snd c)).

Global Program Instance sigT_cofe : Cofe sigTO := { compl := sigT_compl }.
Next Obligation.
intros n c. rewrite /sigT_compl sigT_dist_eq /=.
(symmetry (sigT_chain_const_proj1 c n)).
pose proof (conv_compl n (chain_map_snd c)) as Hgoal.
move: (compl (chain_map_snd c)) Hgoalpc0 /=.
destruct (sigT_chain_const_proj1 c n); simpl. done.
Qed.
End cofe.
End sigT.

Global Arguments sigTO {_} _.

Section sigTOF.
Context {A : Type}.

Program Definition sigT_map {P1 P2 : A ofe} :
discrete_funO (λ a, P1 a -n> P2 a) -n>
sigTO P1 -n> sigTO P2 :=
λne f xpx, existT _ (f _ (projT2 xpx)).
Next Obligation.
move ⇒ ?? f n [x px] [y py] [/= Heq]. destruct Heq; simpl.
eq_refl ⇒ /=. by f_equiv.
Qed.
Next Obligation.
move ⇒ ?? n f g Heq [x px] /=. eq_refl ⇒ /=. apply Heq.
Qed.

Program Definition sigTOF (F : A oFunctor) : oFunctor := {|
oFunctor_car A CA B CB := sigTO (λ a, oFunctor_car (F a) A B);
oFunctor_map A1 _ A2 _ B1 _ B2 _ fg := sigT_map (λ a, oFunctor_map (F a) fg)
|}.
Next Obligation.
repeat intro. eq_refl ⇒ /=. solve_proper.
Qed.
Next Obligation.
simpl; intros. apply (existT_proper eq_refl), oFunctor_map_id.
Qed.
Next Obligation.
simpl; intros. apply (existT_proper eq_refl), oFunctor_map_compose.
Qed.

Global Instance sigTOF_contractive {F} :
( a, oFunctorContractive (F a)) oFunctorContractive (sigTOF F).
Proof.
repeat intro. apply sigT_mapa. exact: oFunctor_map_contractive.
Qed.
End sigTOF.
Global Arguments sigTOF {_} _%OF.

Notation "{ x & P }" := (sigTOF (λ x, P%OF)) : oFunctor_scope.
Notation "{ x : A & P }" := (@sigTOF A%type (λ x, P%OF)) : oFunctor_scope.

# Isomorphisms between OFEs

Record ofe_iso (A B : ofe) := OfeIso {
ofe_iso_1 : A -n> B;
ofe_iso_2 : B -n> A;
ofe_iso_12 y : ofe_iso_1 (ofe_iso_2 y) y;
ofe_iso_21 x : ofe_iso_2 (ofe_iso_1 x) x;
}.
Global Arguments OfeIso {_ _} _ _ _ _.
Global Arguments ofe_iso_1 {_ _} _.
Global Arguments ofe_iso_2 {_ _} _.
Global Arguments ofe_iso_12 {_ _} _ _.
Global Arguments ofe_iso_21 {_ _} _ _.

Section ofe_iso.
Context {A B : ofe}.

Local Instance ofe_iso_equiv : Equiv (ofe_iso A B) := λ I1 I2,
ofe_iso_1 I1 ofe_iso_1 I2 ofe_iso_2 I1 ofe_iso_2 I2.

Local Instance ofe_iso_dist : Dist (ofe_iso A B) := λ n I1 I2,
ofe_iso_1 I1 ≡{n}≡ ofe_iso_1 I2 ofe_iso_2 I1 ≡{n}≡ ofe_iso_2 I2.

Global Instance ofe_iso_1_ne : NonExpansive (ofe_iso_1 (A:=A) (B:=B)).
Proof. by destruct 1. Qed.
Global Instance ofe_iso_2_ne : NonExpansive (ofe_iso_2 (A:=A) (B:=B)).
Proof. by destruct 1. Qed.

Lemma ofe_iso_ofe_mixin : OfeMixin (ofe_iso A B).
Proof. by apply (iso_ofe_mixin (λ I, (ofe_iso_1 I, ofe_iso_2 I))). Qed.
Canonical Structure ofe_isoO : ofe := Ofe (ofe_iso A B) ofe_iso_ofe_mixin.

Global Instance ofe_iso_cofe `{!Cofe A, !Cofe B} : Cofe ofe_isoO.
Proof.
apply (iso_cofe_subtype'
(λ I : prodO (A -n> B) (B -n> A),
( y, I.1 (I.2 y) y) ( x, I.2 (I.1 x) x))
(λ I HI, OfeIso (I.1) (I.2) (proj1 HI) (proj2 HI))
(λ I, (ofe_iso_1 I, ofe_iso_2 I))); [by intros []|done|done|].
apply limit_preserving_and; apply limit_preserving_forall⇒ ?;
apply limit_preserving_equiv; first [intros ???; done|solve_proper].
Qed.
End ofe_iso.

Global Arguments ofe_isoO : clear implicits.

Program Definition iso_ofe_refl {A} : ofe_iso A A := OfeIso cid cid _ _.
Solve Obligations with done.

Definition iso_ofe_sym {A B : ofe} (I : ofe_iso A B) : ofe_iso B A :=
OfeIso (ofe_iso_2 I) (ofe_iso_1 I) (ofe_iso_21 I) (ofe_iso_12 I).
Global Instance iso_ofe_sym_ne {A B} : NonExpansive (iso_ofe_sym (A:=A) (B:=B)).
Proof. intros n I1 I2 []; split; simpl; by f_equiv. Qed.

Program Definition iso_ofe_trans {A B C}
(I : ofe_iso A B) (J : ofe_iso B C) : ofe_iso A C :=
OfeIso (ofe_iso_1 J ofe_iso_1 I) (ofe_iso_2 I ofe_iso_2 J) _ _.
Next Obligation. intros A B C I J z; simpl. by rewrite !ofe_iso_12. Qed.
Next Obligation. intros A B C I J z; simpl. by rewrite !ofe_iso_21. Qed.
Global Instance iso_ofe_trans_ne {A B C} : NonExpansive2 (iso_ofe_trans (A:=A) (B:=B) (C:=C)).
Proof. intros n I1 I2 [] J1 J2 []; split; simpl; by f_equiv. Qed.

Program Definition iso_ofe_cong (F : oFunctor) `{!Cofe A, !Cofe B}
(I : ofe_iso A B) : ofe_iso (oFunctor_apply F A) (oFunctor_apply F B) :=
OfeIso (oFunctor_map F (ofe_iso_2 I, ofe_iso_1 I))
(oFunctor_map F (ofe_iso_1 I, ofe_iso_2 I)) _ _.
Next Obligation.
intros F A ? B ? I x. rewrite -oFunctor_map_compose -{2}(oFunctor_map_id F x).
apply equiv_distn.
apply oFunctor_map_ne; split⇒ ? /=; by rewrite ?ofe_iso_12 ?ofe_iso_21.
Qed.
Next Obligation.
intros F A ? B ? I y. rewrite -oFunctor_map_compose -{2}(oFunctor_map_id F y).
apply equiv_distn.
apply oFunctor_map_ne; split⇒ ? /=; by rewrite ?ofe_iso_12 ?ofe_iso_21.
Qed.
Global Instance iso_ofe_cong_ne (F : oFunctor) `{!Cofe A, !Cofe B} :
NonExpansive (iso_ofe_cong F (A:=A) (B:=B)).
Proof. intros n I1 I2 []; split; simpl; by f_equiv. Qed.
Global Instance iso_ofe_cong_contractive (F : oFunctor) `{!Cofe A, !Cofe B} :
oFunctorContractive F Contractive (iso_ofe_cong F (A:=A) (B:=B)).
Proof. intros ? n I1 I2 HI; split; simpl; f_contractive; by destruct HI. Qed.