Library iris.algebra.ofe

From iris.algebra Require Export base.
Set Default Proof Using "Type".
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.
Unbundled version
Class Dist A := dist : nat relation A.
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).

Hint Extern 0 (_ ≡{_}≡ _) ⇒ reflexivity : core.
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).

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 : x y n, x ≡{n}≡ y;
  mixin_dist_equivalence n : Equivalence (dist n);
  mixin_dist_S n x y : x ≡{S n}≡ y x ≡{n}≡ y
}.

Bundled version
Structure ofeT := OfeT {
  ofe_car :> Type;
  ofe_equiv : Equiv ofe_car;
  ofe_dist : Dist ofe_car;
  ofe_mixin : OfeMixin ofe_car
}.
Arguments OfeT _ {_ _} _.
Add Printing Constructor ofeT.
Hint Extern 0 (Equiv _) ⇒ eapply (@ofe_equiv _) : typeclass_instances.
Hint Extern 0 (Dist _) ⇒ eapply (@ofe_dist _) : typeclass_instances.
Arguments ofe_car : simpl never.
Arguments ofe_equiv : simpl never.
Arguments ofe_dist : simpl never.
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 CmraT and UcmraT 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 @ofe_mixin_of' 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 @ofe_mixin_of' 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 : ofeT} (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 : ofeT}.
  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_S n x y : x ≡{S n}≡ y x ≡{n}≡ y.
  Proof. apply (mixin_dist_S _ (ofe_mixin A)). Qed.
End ofe_mixin.

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

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

Class OfeDiscrete (A : ofeT) := ofe_discrete_discrete (x : A) :> Discrete x.

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

Program Definition chain_map {A B : ofeT} (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 : ofeT) := {
  compl : Compl A;
  conv_compl n c : compl c ≡{n}≡ c n;
}.
Arguments compl : simpl never.
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 : ofeT} (a : A) : chain A :=
  {| chain_car n := a |}.
Next Obligation. by intros A a n i _. Qed.

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

General properties
Section ofe.
  Context {A : ofeT}.
  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. induction 2; eauto using dist_S. Qed.
  Lemma dist_le' n n' x y : n' n x ≡{n}≡ y x ≡{n'}≡ y.
  Proof. intros; eauto using dist_le. Qed.
ne_proper and ne_proper_2 are not instances to improve efficiency of type class search during setoid rewriting. Instances of NonExpansive{,2} are hence accompanied by instances of Proper built using these lemmas.
  Lemma ne_proper {B : ofeT} (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 : ofeT} (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 _), 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.
End ofe.

Contractive functions
Definition dist_later `{Dist A} (n : nat) (x y : A) : Prop :=
  match n with 0 ⇒ True | S nx ≡{n}≡ y end.
Arguments dist_later _ _ !_ _ _ /.

Global Instance dist_later_equivalence (A : ofeT) n : Equivalence (@dist_later A _ n).
Proof. destruct n as [|n]. by split. apply dist_equivalence. Qed.

Lemma dist_dist_later {A : ofeT} n (x y : A) : dist n x y dist_later n x y.
Proof. intros Heq. destruct n; first done. exact: dist_S. Qed.

Lemma dist_later_dist {A : ofeT} n (x y : A) : dist_later (S n) x y dist n x y.
Proof. done. Qed.

Lemma ne_dist_later {A B : ofeT} (f : A B) :
  NonExpansive f n, Proper (dist_later n ==> dist_later n) f.
Proof. intros Hf [|n]; last exact: Hf. hnf. by intros. Qed.

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

Instance const_contractive {A B : ofeT} (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 : ofeT} (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). Qed.
  Lemma contractive_S n x y : x ≡{n}≡ y f x ≡{S n}≡ f y.
  Proof. intros. by apply (_ : Contractive f). Qed.

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

Ltac f_contractive :=
  match goal with
  | |- ?f _ ≡{_}≡ ?f _simple apply (_ : Proper (dist_later _ ==> _) f)
  | |- ?f _ _ ≡{_}≡ ?f _ _simple apply (_ : Proper (dist_later _ ==> _ ==> _) f)
  | |- ?f _ _ ≡{_}≡ ?f _ _simple apply (_ : Proper (_ ==> dist_later _ ==> _) f)
  end;
  try match goal with
  | |- @dist_later ?A _ ?n ?x ?y
         destruct n as [|n]; [exact I|change (@dist A _ n x y)]
  end;
  try simple apply 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).
Hint Mode LimitPreserving + + ! : typeclass_instances.

Section limit_preserving.
  Context `{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.

  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 : ofeT} `{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.

Program Definition fixpoint_def `{Cofe A, Inhabited A} (f : A A)
  `{!Contractive f} : A := compl (fixpoint_chain f).
Definition fixpoint_aux : seal (@fixpoint_def). Proof. by eexists. Qed.
Definition fixpoint {A AC AiH} f {Hf} := fixpoint_aux.(unseal) A AC AiH f Hf.
Definition fixpoint_eq : @fixpoint = @fixpoint_def := fixpoint_aux.(seal_eq).

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

  Lemma fixpoint_unfold : fixpoint f f (fixpoint f).
  Proof.
    apply equiv_distn.
    rewrite fixpoint_eq /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. apply (contractive_S _), IH.
  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_eq /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; auto using dist_S.
  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 lia. }
    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 `{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 `{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_S_r Nat_iter_S -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.
  Local Unset Default Proof Using.

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

  Instance: Proper ((≡) ==> (≡) ==> (≡)) fA.
  Proof.
    apply ne_proper_2n x x' ? y y' ?. f_contractive; auto using dist_S.
  Qed.
  Instance: Proper ((≡) ==> (≡) ==> (≡)) fB.
  Proof.
    apply ne_proper_2n x x' ? y y' ?. f_contractive; auto using dist_S.
  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 `{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; auto using dist_S.
  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 : ofeT) : Type := OfeMor {
  ofe_mor_car :> A B;
  ofe_mor_ne : NonExpansive ofe_mor_car
}.
Arguments OfeMor {_ _} _ {_}.
Add Printing Constructor ofe_mor.
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 : ofeT}.
  Global Instance ofe_mor_proper (f : ofe_mor A B) : Proper ((≡) ==> (≡)) f.
  Proof. apply ne_proper, ofe_mor_ne. Qed.
  Instance ofe_mor_equiv : Equiv (ofe_mor A B) := λ f g, x, f x g x.
  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).
    - by intros n f g ? x; apply dist_S.
  Qed.
  Canonical Structure ofe_morO := OfeT (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.

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

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

Definition ccompose {A B C}
  (f : B -n> C) (g : A -n> B) : A -n> C := OfeMor (f g).
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.

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.
Instance ofe_mor_map_ne {A A' B B'} n :
  Proper (dist n ==> dist n ==> dist n ==> dist n) (@ofe_mor_map A A' B B').
Proof. intros ??? ??? ???. 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).
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.
  Instance unit_dist : Dist unit := λ _ _ _, True.
  Definition unit_ofe_mixin : OfeMixin unit.
  Proof. by repeat split; try 0. Qed.
  Canonical Structure unitO : ofeT := OfeT 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.
  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 : ofeT := OfeT 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 : ofeT}.

  Instance prod_dist : Dist (A × B) := λ n, prod_relation (dist n) (dist n).
  Global Instance pair_ne :
    NonExpansive2 (@pair A B) := _.
  Global Instance fst_ne : NonExpansive (@fst A B) := _.
  Global Instance snd_ne : NonExpansive (@snd A B) := _.
  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 [x1 y1] [x2 y2] [??]; split; apply dist_S.
  Qed.
  Canonical Structure prodO : ofeT := OfeT (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 _). Qed.
  Global Instance prod_ofe_discrete :
    OfeDiscrete A OfeDiscrete B OfeDiscrete prodO.
  Proof. intros ?? [??]; apply _. Qed.
End product.

Arguments prodO : clear implicits.
Typeclasses Opaque prod_dist.

Instance prod_map_ne {A A' B B' : ofeT} 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).
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}, ofeT;
  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)
}.
Existing Instance oFunctor_map_ne.
Instance: Params (@oFunctor_map) 9 := {}.

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

Class oFunctorContractive (F : oFunctor) :=
  oFunctor_map_contractive `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} :>
    Contractive (@oFunctor_map F A1 _ A2 _ B1 _ B2 _).
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: ofeT) `{!Cofe A} : ofeT :=
  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.
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.
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 : ofeT) : oFunctor :=
  {| oFunctor_car A1 A2 _ _ := B; oFunctor_map A1 _ A2 _ B1 _ B2 _ f := cid |}.
Solve Obligations with done.
Coercion constOF : ofeT >-> oFunctor.

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.

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.

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; destruct n, Hfg; by split.
Qed.

Sum type

Section sum.
  Context {A B : ofeT}.

  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 : Inj (dist n) (dist n) (@inl A B) := _.
  Global Instance inr_ne_inj : 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; by apply dist_S.
  Qed.
  Canonical Structure sumO : ofeT := OfeT (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.
    feed inversion (chain_cauchy c 0 n); first by auto with lia; constructor.
    - 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 _). Qed.
  Global Instance inr_discrete (y : B) : Discrete y Discrete (inr y).
  Proof. inversion_clear 2; constructor; by apply (discrete _). Qed.
  Global Instance sum_ofe_discrete :
    OfeDiscrete A OfeDiscrete B OfeDiscrete sumO.
  Proof. intros ?? [?|?]; apply _. Qed.
End sum.

Arguments sumO : clear implicits.
Typeclasses Opaque sum_dist.

Instance sum_map_ne {A A' B B' : ofeT} 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).
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.

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 `{Equiv A} (Heq : @Equivalence A (≡)).

  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 (OfeT A discrete_ofe_mixin).
  Proof. by intros x y. Qed.

  Global Program Instance discrete_cofe : Cofe (OfeT 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.

Notation discreteO A := (OfeT A (discrete_ofe_mixin _)).
Force the Equivalence proof to be eq_equivalence so that it does not find 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).

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

Option type

Section option.
  Context {A : ofeT}.

  Instance option_dist : Dist (option A) := λ n, option_Forall2 (dist n).
  Lemma dist_option_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; feed inversion (Hxy n).
    - apply _.
    - destruct 1; constructor; by apply dist_S.
  Qed.
  Canonical Structure optionO := OfeT (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.
    feed inversion (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 _). 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 : Inj (dist n) (dist n) (@Some A).
  Proof. by inversion_clear 1. Qed.
  Global Instance from_option_ne {B} (R : relation B) (f : A B) n :
    Proper (dist n ==> R) f Proper (R ==> dist n ==> R) (from_option f).
  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. 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.

Typeclasses Opaque option_dist.
Arguments optionO : clear implicits.

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

Definition optionO_map {A B} (f : A -n> B) : optionO A -n> optionO B :=
  OfeMor (fmap f : optionO A optionO B).
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.

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.
Arguments Next {_} _.
Arguments later_car {_} _.
Instance: Params (@Next) 1 := {}.

Section later.
  Context {A : ofeT}.
  Instance later_equiv : Equiv (later A) := λ x y, later_car x later_car y.
  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|apply Hxy]. intros Hxy n; apply (Hxy (S n)).
    - split; rewrite /dist /later_dist.
      + by intros [x].
      + by intros [x] [y].
      + by intros [x] [y] [z] ??; trans y.
    - intros [|n] [x] [y] ?; [done|]; rewrite /dist /later_dist; by apply dist_S.
  Qed.
  Canonical Structure laterO : ofeT := OfeT (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; [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 Later_inj n : Inj (dist n) (dist (S n)) (@Next A).
  Proof. by intros x y. Qed.

  Lemma Next_uninj x : a, x Next a.
  Proof. by (later_car x). Qed.
  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 : ofeT} (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.

Arguments laterO : clear implicits.

Definition later_map {A B} (f : A B) (x : later A) : later B :=
  Next (f (later_car x)).
Instance later_map_ne {A B : ofeT} (f : A B) n :
  Proper (dist (pred n) ==> dist (pred n)) f
  Proper (dist n ==> dist n) (later_map f) | 0.
Proof. destruct n as [|n]; intros Hf [x] [y] ?; do 2 red; simpl; auto. Qed.
Instance later_map_proper {A B : ofeT} (f : A B) :
  Proper ((≡) ==> (≡)) f
  Proper ((≡) ==> (≡)) (later_map f).
Proof. solve_proper. 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 : ofeT} (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).
Instance laterO_map_contractive (A B : ofeT) : Contractive (@laterO_map A B).
Proof. intros [|n] f g Hf n'; [done|]; apply Hf; lia. 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.

Instance laterOF_contractive F : oFunctorContractive (laterOF F).
Proof.
  intros A1 ? A2 ? B1 ? B2 ? n fg fg' Hfg. apply laterO_map_contractive.
  destruct n as [|n]; simpl in *; first done. apply 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 ofeT) := x : A, B x.

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

  Instance discrete_fun_equiv : Equiv (discrete_fun B) := λ f g, x, f x g x.
  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 f g ? x; apply dist_S.
  Qed.
  Canonical Structure discrete_funO := OfeT (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.

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 ofeT} (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 ofeT} (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 ofeT} (g : discrete_fun B) :
  discrete_fun_map (λ _, id) g = g.
Proof. done. Qed.
Lemma discrete_fun_map_compose {A} {B1 B2 B3 : A ofeT}
    (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.

Instance discrete_fun_map_ne {A} {B1 B2 : A ofeT} (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 ofeT} (f : discrete_fun (λ x, B1 x -n> B2 x)) :
  discrete_funO B1 -n> discrete_funO B2 := OfeMor (discrete_fun_map f).
Instance discrete_funO_map_ne {A} {B1 B2 : A ofeT} :
  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.

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 : ofeT} `{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 y1 y2. rewrite !g_dist. apply dist_S.
Qed.

Section iso_cofe_subtype.
  Context {A B : ofeT} `{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. Qed.
  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 : ofeT} `{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 : ofeT} `{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 : ofeT} {P : A Prop}.
  Implicit Types x : sig P.

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

  Definition sig_equiv_alt x y : x y `x `y := reflexivity _.
  Definition sig_dist_alt 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 : ofeT := OfeT (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_alt sig_equiv_alt. apply (discrete _). Qed.
  Global Instance sig_ofe_discrete : OfeDiscrete A OfeDiscrete sigO.
  Proof. intros ??. apply _. Qed.
End sigma.

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 ofeT}.
  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.
  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.
  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.
      move ⇒ [xa x] [ya y] /=. destruct 1 as [-> Heq].
       eq_refl. exact: dist_S.
  Qed.

  Canonical Structure sigTO : ofeT := OfeT (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 _), 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.

Arguments sigTO {_} _.

Section sigTOF.
  Context {A : Type}.

  Program Definition sigT_map {P1 P2 : A ofeT} :
    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.
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 : ofeT) := 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;
}.
Arguments OfeIso {_ _} _ _ _ _.
Arguments ofe_iso_1 {_ _} _.
Arguments ofe_iso_2 {_ _} _.
Arguments ofe_iso_12 {_ _} _ _.
Arguments ofe_iso_21 {_ _} _ _.

Section ofe_iso.
  Context {A B : ofeT}.

  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.

  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 : ofeT := OfeT (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.

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 : ofeT} (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).
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.
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.
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.
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.