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From iris.algebra Require Export cmra.
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From stdpp Require Export list.
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From iris.base_logic Require Import base_logic.
From iris.algebra Require Import updates local_updates.
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Set Default Proof Using "Type".
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Section cofe.
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Context {A : ofeT}.
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Implicit Types l : list A.
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Instance list_dist : Dist (list A) := λ n, Forall2 (dist n).

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Lemma list_dist_lookup n l1 l2 : l1 {n} l2   i, l1 !! i {n} l2 !! i.
Proof. setoid_rewrite dist_option_Forall2. apply Forall2_lookup. Qed.

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Global Instance cons_ne : NonExpansive2 (@cons A) := _.
Global Instance app_ne : NonExpansive2 (@app A) := _.
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Global Instance length_ne n : Proper (dist n ==> (=)) (@length A) := _.
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Global Instance tail_ne : NonExpansive (@tail A) := _.
Global Instance take_ne : NonExpansive (@take A n) := _.
Global Instance drop_ne : NonExpansive (@drop A n) := _.
Global Instance list_lookup_ne i :
  NonExpansive (lookup (M:=list A) i).
Proof. intros ????. by apply dist_option_Forall2, Forall2_lookup. Qed.
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Global Instance list_alter_ne n f i :
  Proper (dist n ==> dist n) f 
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  Proper (dist n ==> dist n) (alter (M:=list A) f i) := _.
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Global Instance list_insert_ne i :
  NonExpansive2 (insert (M:=list A) i) := _.
Global Instance list_inserts_ne i :
  NonExpansive2 (@list_inserts A i) := _.
Global Instance list_delete_ne i :
  NonExpansive (delete (M:=list A) i) := _.
Global Instance option_list_ne : NonExpansive (@option_list A).
Proof. intros ????; by apply Forall2_option_list, dist_option_Forall2. Qed.
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Global Instance list_filter_ne n P `{ x, Decision (P x)} :
  Proper (dist n ==> iff) P 
  Proper (dist n ==> dist n) (filter (B:=list A) P) := _.
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Global Instance replicate_ne :
  NonExpansive (@replicate A n) := _.
Global Instance reverse_ne : NonExpansive (@reverse A) := _.
Global Instance last_ne : NonExpansive (@last A).
Proof. intros ????; by apply dist_option_Forall2, Forall2_last. Qed.
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Global Instance resize_ne n :
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  NonExpansive2 (@resize A n) := _.
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Definition list_ofe_mixin : OfeMixin (list A).
Proof.
  split.
  - intros l k. rewrite equiv_Forall2 -Forall2_forall.
    split; induction 1; constructor; intros; try apply equiv_dist; auto.
  - apply _.
  - rewrite /dist /list_dist. eauto using Forall2_impl, dist_S.
Qed.
Canonical Structure listC := OfeT (list A) list_ofe_mixin.

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Program Definition list_chain
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    (c : chain listC) (x : A) (k : nat) : chain A :=
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  {| chain_car n := from_option id x (c n !! k) |}.
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Next Obligation. intros c x k n i ?. by rewrite /= (chain_cauchy c n i). Qed.
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Definition list_compl `{Cofe A} : Compl listC := λ c,
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  match c 0 with
  | [] => []
  | x :: _ => compl  list_chain c x <$> seq 0 (length (c 0))
  end.
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Global Program Instance list_cofe `{Cofe A} : Cofe listC :=
  {| compl := list_compl |}.
Next Obligation.
  intros ? n c; rewrite /compl /list_compl.
  destruct (c 0) as [|x l] eqn:Hc0 at 1.
  { by destruct (chain_cauchy c 0 n); auto with omega. }
  rewrite -(λ H, length_ne _ _ _ (chain_cauchy c 0 n H)); last omega.
  apply Forall2_lookup=> i. rewrite -dist_option_Forall2 list_lookup_fmap.
  destruct (decide (i < length (c n))); last first.
  { rewrite lookup_seq_ge ?lookup_ge_None_2; auto with omega. }
  rewrite lookup_seq //= (conv_compl n (list_chain c _ _)) /=.
  destruct (lookup_lt_is_Some_2 (c n) i) as [? Hcn]; first done.
  by rewrite Hcn.
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Qed.
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Global Instance list_ofe_discrete : OfeDiscrete A  OfeDiscrete listC.
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Proof. induction 2; constructor; try apply (discrete _); auto. Qed.
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Global Instance nil_discrete : Discrete (@nil A).
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Proof. inversion_clear 1; constructor. Qed.
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Global Instance cons_discrete x l : Discrete x  Discrete l  Discrete (x :: l).
Proof. intros ??; inversion_clear 1; constructor; by apply discrete. Qed.
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End cofe.

Arguments listC : clear implicits.

(** Functor *)
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Lemma list_fmap_ext_ne {A} {B : ofeT} (f g : A  B) (l : list A) n :
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  ( x, f x {n} g x)  f <$> l {n} g <$> l.
Proof. intros Hf. by apply Forall2_fmap, Forall_Forall2, Forall_true. Qed.
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Instance list_fmap_ne {A B : ofeT} (f : A  B) n:
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  Proper (dist n ==> dist n) f  Proper (dist n ==> dist n) (fmap (M:=list) f).
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Proof. intros Hf l k ?; by eapply Forall2_fmap, Forall2_impl; eauto. Qed.
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Definition listC_map {A B} (f : A -n> B) : listC A -n> listC B :=
  CofeMor (fmap f : listC A  listC B).
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Instance listC_map_ne A B : NonExpansive (@listC_map A B).
Proof. intros n f g ? l. by apply list_fmap_ext_ne. Qed.
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Program Definition listCF (F : cFunctor) : cFunctor := {|
  cFunctor_car A B := listC (cFunctor_car F A B);
  cFunctor_map A1 A2 B1 B2 fg := listC_map (cFunctor_map F fg)
|}.
Next Obligation.
  by intros F A1 A2 B1 B2 n f g Hfg; apply listC_map_ne, cFunctor_ne.
Qed.
Next Obligation.
  intros F A B x. rewrite /= -{2}(list_fmap_id x).
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  apply list_fmap_equiv_ext=>y. apply cFunctor_id.
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Qed.
Next Obligation.
  intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -list_fmap_compose.
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  apply list_fmap_equiv_ext=>y; apply cFunctor_compose.
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Qed.

Instance listCF_contractive F :
  cFunctorContractive F  cFunctorContractive (listCF F).
Proof.
  by intros ? A1 A2 B1 B2 n f g Hfg; apply listC_map_ne, cFunctor_contractive.
Qed.
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(* CMRA *)
Section cmra.
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  Context {A : ucmraT}.
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  Implicit Types l : list A.
  Local Arguments op _ _ !_ !_ / : simpl nomatch.

  Instance list_op : Op (list A) :=
    fix go l1 l2 := let _ : Op _ := @go in
    match l1, l2 with
    | [], _ => l2
    | _, [] => l1
    | x :: l1, y :: l2 => x  y :: l1  l2
    end.
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  Instance list_pcore : PCore (list A) := λ l, Some (core <$> l).
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  Instance list_valid : Valid (list A) := Forall (λ x,  x).
  Instance list_validN : ValidN (list A) := λ n, Forall (λ x, {n} x).

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  Lemma cons_valid l x :  (x :: l)   x   l.
  Proof. apply Forall_cons. Qed.
  Lemma cons_validN n l x : {n} (x :: l)  {n} x  {n} l.
  Proof. apply Forall_cons. Qed.
  Lemma app_valid l1 l2 :  (l1 ++ l2)   l1   l2.
  Proof. apply Forall_app. Qed.
  Lemma app_validN n l1 l2 : {n} (l1 ++ l2)  {n} l1  {n} l2.
  Proof. apply Forall_app. Qed.

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  Lemma list_lookup_valid l :  l   i,  (l !! i).
  Proof.
    rewrite {1}/valid /list_valid Forall_lookup; split.
    - intros Hl i. by destruct (l !! i) as [x|] eqn:?; [apply (Hl i)|].
    - intros Hl i x Hi. move: (Hl i); by rewrite Hi.
  Qed.
  Lemma list_lookup_validN n l : {n} l   i, {n} (l !! i).
  Proof.
    rewrite {1}/validN /list_validN Forall_lookup; split.
    - intros Hl i. by destruct (l !! i) as [x|] eqn:?; [apply (Hl i)|].
    - intros Hl i x Hi. move: (Hl i); by rewrite Hi.
  Qed.
  Lemma list_lookup_op l1 l2 i : (l1  l2) !! i = l1 !! i  l2 !! i.
  Proof.
    revert i l2. induction l1 as [|x l1]; intros [|i] [|y l2];
      by rewrite /= ?left_id_L ?right_id_L.
  Qed.
  Lemma list_lookup_core l i : core l !! i = core (l !! i).
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  Proof.
    rewrite /core /= list_lookup_fmap.
    destruct (l !! i); by rewrite /= ?Some_core.
  Qed.
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  Lemma list_lookup_included l1 l2 : l1  l2   i, l1 !! i  l2 !! i.
  Proof.
    split.
    { intros [l Hl] i. exists (l !! i). by rewrite Hl list_lookup_op. }
    revert l1. induction l2 as [|y l2 IH]=>-[|x l1] Hl.
    - by exists [].
    - destruct (Hl 0) as [[z|] Hz]; inversion Hz.
    - by exists (y :: l2).
    - destruct (IH l1) as [l3 ?]; first (intros i; apply (Hl (S i))).
      destruct (Hl 0) as [[z|] Hz]; inversion_clear Hz; simplify_eq/=.
      + exists (z :: l3); by constructor.
      + exists (core x :: l3); constructor; by rewrite ?cmra_core_r.
  Qed.

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  Definition list_cmra_mixin : CmraMixin (list A).
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  Proof.
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    apply cmra_total_mixin.
    - eauto.
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    - intros n l l1 l2; rewrite !list_dist_lookup=> Hl i.
      by rewrite !list_lookup_op Hl.
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    - intros n l1 l2 Hl; by rewrite /core /= Hl.
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    - intros n l1 l2; rewrite !list_dist_lookup !list_lookup_validN=> Hl ? i.
      by rewrite -Hl.
    - intros l. rewrite list_lookup_valid. setoid_rewrite list_lookup_validN.
      setoid_rewrite cmra_valid_validN. naive_solver.
    - intros n x. rewrite !list_lookup_validN. auto using cmra_validN_S.
    - intros l1 l2 l3; rewrite list_equiv_lookup=> i.
      by rewrite !list_lookup_op assoc.
    - intros l1 l2; rewrite list_equiv_lookup=> i.
      by rewrite !list_lookup_op comm.
    - intros l; rewrite list_equiv_lookup=> i.
      by rewrite list_lookup_op list_lookup_core cmra_core_l.
    - intros l; rewrite list_equiv_lookup=> i.
      by rewrite !list_lookup_core cmra_core_idemp.
    - intros l1 l2; rewrite !list_lookup_included=> Hl i.
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      rewrite !list_lookup_core. by apply cmra_core_mono.
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    - intros n l1 l2. rewrite !list_lookup_validN.
      setoid_rewrite list_lookup_op. eauto using cmra_validN_op_l.
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    - intros n l.
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      induction l as [|x l IH]=> -[|y1 l1] [|y2 l2] Hl Heq;
        (try by exfalso; inversion Heq).
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      + by exists [], [].
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      + exists [], (x :: l); inversion Heq; by repeat constructor.
      + exists (x :: l), []; inversion Heq; by repeat constructor.
      + destruct (IH l1 l2) as (l1'&l2'&?&?&?),
          (cmra_extend n x y1 y2) as (y1'&y2'&?&?&?);
          [by inversion_clear Heq; inversion_clear Hl..|].
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        exists (y1' :: l1'), (y2' :: l2'); repeat constructor; auto.
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  Qed.
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  Canonical Structure listR := CmraT (list A) list_cmra_mixin.
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  Global Instance list_unit : Unit (list A) := [].
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  Definition list_ucmra_mixin : UcmraMixin (list A).
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  Proof.
    split.
    - constructor.
    - by intros l.
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    - by constructor.
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  Qed.
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  Canonical Structure listUR := UcmraT (list A) list_ucmra_mixin.
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  Global Instance list_cmra_discrete : CmraDiscrete A  CmraDiscrete listR.
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  Proof.
    split; [apply _|]=> l; rewrite list_lookup_valid list_lookup_validN=> Hl i.
    by apply cmra_discrete_valid.
  Qed.

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  Global Instance list_core_id l : ( x : A, CoreId x)  CoreId l.
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  Proof.
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    intros ?; constructor; apply list_equiv_lookup=> i.
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    by rewrite list_lookup_core (core_id_core (l !! i)).
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  Qed.

  (** Internalized properties *)
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  Lemma list_equivI {M} l1 l2 : l1  l2  ( i, l1 !! i  l2 !! i : uPred M).
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  Proof. uPred.unseal; constructor=> n x ?. apply list_dist_lookup. Qed.
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  Lemma list_validI {M} l :  l  ( i,  (l !! i) : uPred M).
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  Proof. uPred.unseal; constructor=> n x ?. apply list_lookup_validN. Qed.
End cmra.

Arguments listR : clear implicits.
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Arguments listUR : clear implicits.
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Instance list_singletonM {A : ucmraT} : SingletonM nat A (list A) := λ n x,
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  replicate n ε ++ [x].
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Section properties.
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  Context {A : ucmraT}.
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  Implicit Types l : list A.
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  Implicit Types x y z : A.
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  Local Arguments op _ _ !_ !_ / : simpl nomatch.
  Local Arguments cmra_op _ !_ !_ / : simpl nomatch.
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  Local Arguments ucmra_op _ !_ !_ / : simpl nomatch.
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  Lemma list_lookup_opM l mk i : (l ? mk) !! i = l !! i  (mk = (!! i)).
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  Proof. destruct mk; by rewrite /= ?list_lookup_op ?right_id_L. Qed.

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  Global Instance list_op_nil_l : LeftId (=) (@nil A) op.
  Proof. done. Qed.
  Global Instance list_op_nil_r : RightId (=) (@nil A) op.
  Proof. by intros []. Qed.

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  Lemma list_op_app l1 l2 l3 :
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    (l1 ++ l3)  l2 = (l1  take (length l1) l2) ++ (l3  drop (length l1) l2).
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  Proof.
    revert l2 l3.
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    induction l1 as [|x1 l1]=> -[|x2 l2] [|x3 l3]; f_equal/=; auto.
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  Qed.
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  Lemma list_op_app_le l1 l2 l3 :
    length l2  length l1  (l1 ++ l3)  l2 = (l1  l2) ++ l3.
  Proof. intros ?. by rewrite list_op_app take_ge // drop_ge // right_id_L. Qed.
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  Lemma list_lookup_validN_Some n l i x : {n} l  l !! i {n} Some x  {n} x.
  Proof. move=> /list_lookup_validN /(_ i)=> Hl Hi; move: Hl. by rewrite Hi. Qed.
  Lemma list_lookup_valid_Some l i x :  l  l !! i  Some x   x.
  Proof. move=> /list_lookup_valid /(_ i)=> Hl Hi; move: Hl. by rewrite Hi. Qed.

  Lemma list_op_length l1 l2 : length (l1  l2) = max (length l1) (length l2).
  Proof. revert l2. induction l1; intros [|??]; f_equal/=; auto. Qed.

  Lemma replicate_valid n (x : A) :  x   replicate n x.
  Proof. apply Forall_replicate. Qed.
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  Global Instance list_singletonM_ne i :
    NonExpansive (@list_singletonM A i).
  Proof. intros n l1 l2 ?. apply Forall2_app; by repeat constructor. Qed.
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  Global Instance list_singletonM_proper i :
    Proper (() ==> ()) (list_singletonM i) := ne_proper _.
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  Lemma elem_of_list_singletonM i z x : z  ({[i := x]} : list A)  z = ε  z = x.
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  Proof.
    rewrite elem_of_app elem_of_list_singleton elem_of_replicate. naive_solver.
  Qed.
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  Lemma list_lookup_singletonM i x : ({[ i := x ]} : list A) !! i = Some x.
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  Proof. induction i; by f_equal/=. Qed.
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  Lemma list_lookup_singletonM_ne i j x :
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    i  j 
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    ({[ i := x ]} : list A) !! j = None  ({[ i := x ]} : list A) !! j = Some ε.
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  Proof. revert j; induction i; intros [|j]; naive_solver auto with omega. Qed.
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  Lemma list_singletonM_validN n i x : {n} ({[ i := x ]} : list A)  {n} x.
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  Proof.
    rewrite list_lookup_validN. split.
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    { move=> /(_ i). by rewrite list_lookup_singletonM. }
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    intros Hx j; destruct (decide (i = j)); subst.
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    - by rewrite list_lookup_singletonM.
    - destruct (list_lookup_singletonM_ne i j x) as [Hi|Hi]; first done;
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        rewrite Hi; by try apply (ucmra_unit_validN (A:=A)).
  Qed.
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  Lemma list_singleton_valid  i x :  ({[ i := x ]} : list A)   x.
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  Proof.
    rewrite !cmra_valid_validN. by setoid_rewrite list_singletonM_validN.
  Qed.
  Lemma list_singletonM_length i x : length {[ i := x ]} = S i.
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  Proof.
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    rewrite /singletonM /list_singletonM app_length replicate_length /=; lia.
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  Qed.

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  Lemma list_core_singletonM i (x : A) : core {[ i := x ]}  {[ i := core x ]}.
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  Proof.
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    rewrite /singletonM /list_singletonM.
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    by rewrite {1}/core /= fmap_app fmap_replicate (core_id_core ).
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  Qed.
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  Lemma list_op_singletonM i (x y : A) :
    {[ i := x ]}  {[ i := y ]}  {[ i := x  y ]}.
  Proof.
    rewrite /singletonM /list_singletonM /=.
    induction i; constructor; rewrite ?left_id; auto.
  Qed.
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  Lemma list_alter_singletonM f i x :
    alter f i ({[i := x]} : list A) = {[i := f x]}.
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  Proof.
    rewrite /singletonM /list_singletonM /=. induction i; f_equal/=; auto.
  Qed.
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  Global Instance list_singleton_core_id i (x : A) :
    CoreId x  CoreId {[ i := x ]}.
  Proof. by rewrite !core_id_total list_core_singletonM=> ->. Qed.
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  (* Update *)
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  Lemma list_singleton_updateP (P : A  Prop) (Q : list A  Prop) x :
    x ~~>: P  ( y, P y  Q [y])  [x] ~~>: Q.
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  Proof.
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    rewrite !cmra_total_updateP=> Hup HQ n lf /list_lookup_validN Hv.
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    destruct (Hup n (from_option id ε (lf !! 0))) as (y&?&Hv').
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    { move: (Hv 0). by destruct lf; rewrite /= ?right_id. }
    exists [y]; split; first by auto.
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    apply list_lookup_validN=> i.
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    move: (Hv i) Hv'. by destruct i, lf; rewrite /= ?right_id.
  Qed.
  Lemma list_singleton_updateP' (P : A  Prop) x :
    x ~~>: P  [x] ~~>: λ k,  y, k = [y]  P y.
  Proof. eauto using list_singleton_updateP. Qed.
  Lemma list_singleton_update x y : x ~~> y  [x] ~~> [y].
  Proof.
    rewrite !cmra_update_updateP; eauto using list_singleton_updateP with subst.
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  Qed.

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  Lemma app_updateP (P1 P2 Q : list A  Prop) l1 l2 :
    l1 ~~>: P1  l2 ~~>: P2 
    ( k1 k2, P1 k1  P2 k2  length l1 = length k1  Q (k1 ++ k2)) 
    l1 ++ l2 ~~>: Q.
  Proof.
    rewrite !cmra_total_updateP=> Hup1 Hup2 HQ n lf.
    rewrite list_op_app app_validN=> -[??].
    destruct (Hup1 n (take (length l1) lf)) as (k1&?&?); auto.
    destruct (Hup2 n (drop (length l1) lf)) as (k2&?&?); auto.
    exists (k1 ++ k2). rewrite list_op_app app_validN.
    by destruct (HQ k1 k2) as [<- ?].
  Qed.
  Lemma app_update l1 l2 k1 k2 :
    length l1 = length k1 
    l1 ~~> k1  l2 ~~> k2  l1 ++ l2 ~~> k1 ++ k2.
  Proof. rewrite !cmra_update_updateP; eauto using app_updateP with subst. Qed.

  Lemma cons_updateP (P1 : A  Prop) (P2 Q : list A  Prop) x l :
    x ~~>: P1  l ~~>: P2  ( y k, P1 y  P2 k  Q (y :: k))  x :: l ~~>: Q.
  Proof.
    intros. eapply (app_updateP _ _ _ [x]);
      naive_solver eauto using list_singleton_updateP'.
  Qed.
  Lemma cons_updateP' (P1 : A  Prop) (P2 : list A  Prop) x l :
    x ~~>: P1  l ~~>: P2  x :: l ~~>: λ k,  y k', k = y :: k'  P1 y  P2 k'.
  Proof. eauto 10 using cons_updateP. Qed.
  Lemma cons_update x y l k : x ~~> y  l ~~> k  x :: l ~~> y :: k.
  Proof. rewrite !cmra_update_updateP; eauto using cons_updateP with subst. Qed.

  Lemma list_middle_updateP (P : A  Prop) (Q : list A  Prop) l1 x l2 :
    x ~~>: P  ( y, P y  Q (l1 ++ y :: l2))  l1 ++ x :: l2 ~~>: Q.
  Proof.
    intros. eapply app_updateP.
    - by apply cmra_update_updateP.
    - by eapply cons_updateP', cmra_update_updateP.
    - naive_solver.
  Qed.
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  Lemma list_middle_update l1 l2 x y : x ~~> y  l1 ++ x :: l2 ~~> l1 ++ y :: l2.
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  Proof.
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    rewrite !cmra_update_updateP=> ?; eauto using list_middle_updateP with subst.
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  Qed.

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(* FIXME
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  Lemma list_middle_local_update l1 l2 x y ml :
    x ~l~> y @ ml ≫= (!! length l1) →
    l1 ++ x :: l2 ~l~> l1 ++ y :: l2 @ ml.
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  Proof.
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    intros [Hxy Hxy']; split.
    - intros n; rewrite !list_lookup_validN=> Hl i; move: (Hl i).
      destruct (lt_eq_lt_dec i (length l1)) as [[?|?]|?]; subst.
      + by rewrite !list_lookup_opM !lookup_app_l.
      + rewrite !list_lookup_opM !list_lookup_middle // !Some_op_opM; apply (Hxy n).
      + rewrite !(cons_middle _ l1 l2) !assoc.
        rewrite !list_lookup_opM !lookup_app_r !app_length //=; lia.
    - intros n mk; rewrite !list_lookup_validN !list_dist_lookup => Hl Hl' i.
      move: (Hl i) (Hl' i).
      destruct (lt_eq_lt_dec i (length l1)) as [[?|?]|?]; subst.
      + by rewrite !list_lookup_opM !lookup_app_l.
      + rewrite !list_lookup_opM !list_lookup_middle // !Some_op_opM !inj_iff.
        apply (Hxy' n).
      + rewrite !(cons_middle _ l1 l2) !assoc.
        rewrite !list_lookup_opM !lookup_app_r !app_length //=; lia.
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  Qed.
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  Lemma list_singleton_local_update i x y ml :
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    x ~l~> y @ ml ≫= (!! i) → {[ i := x ]} ~l~> {[ i := y ]} @ ml.
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  Proof. intros; apply list_middle_local_update. by rewrite replicate_length. Qed.
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*)
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End properties.

(** Functor *)
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Instance list_fmap_cmra_morphism {A B : ucmraT} (f : A  B)
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  `{!CmraMorphism f} : CmraMorphism (fmap f : list A  list B).
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Proof.
  split; try apply _.
  - intros n l. rewrite !list_lookup_validN=> Hl i. rewrite list_lookup_fmap.
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    by apply (cmra_morphism_validN (fmap f : option A  option B)).
  - intros l. apply Some_proper. rewrite -!list_fmap_compose.
    apply list_fmap_equiv_ext, cmra_morphism_core, _.
  - intros l1 l2. apply list_equiv_lookup=>i.
    by rewrite list_lookup_op !list_lookup_fmap list_lookup_op cmra_morphism_op.
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Qed.

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Program Definition listURF (F : urFunctor) : urFunctor := {|
  urFunctor_car A B := listUR (urFunctor_car F A B);
  urFunctor_map A1 A2 B1 B2 fg := listC_map (urFunctor_map F fg)
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|}.
Next Obligation.
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  by intros F ???? n f g Hfg; apply listC_map_ne, urFunctor_ne.
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Qed.
Next Obligation.
  intros F A B x. rewrite /= -{2}(list_fmap_id x).
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  apply list_fmap_equiv_ext=>y. apply urFunctor_id.
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Qed.
Next Obligation.
  intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -list_fmap_compose.
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  apply list_fmap_equiv_ext=>y; apply urFunctor_compose.
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Qed.

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Instance listURF_contractive F :
  urFunctorContractive F  urFunctorContractive (listURF F).
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Proof.
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  by intros ? A1 A2 B1 B2 n f g Hfg; apply listC_map_ne, urFunctor_contractive.
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Qed.