Forked from
Iris / Iris
3954 commits behind the upstream repository.
list.v 19.10 KiB
From iris.algebra Require Export cmra.
From stdpp Require Export list.
From iris.base_logic Require Import base_logic.
From iris.algebra Require Import updates local_updates.
Set Default Proof Using "Type".
Section cofe.
Context {A : ofeT}.
Implicit Types l : list A.
Instance list_dist : Dist (list A) := λ n, Forall2 (dist n).
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.
Global Instance cons_ne : NonExpansive2 (@cons A) := _.
Global Instance app_ne : NonExpansive2 (@app A) := _.
Global Instance length_ne n : Proper (dist n ==> (=)) (@length A) := _.
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.
Global Instance list_alter_ne n f i :
Proper (dist n ==> dist n) f →
Proper (dist n ==> dist n) (alter (M:=list A) f i) := _.
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.
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) := _.
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.
Global Instance resize_ne n :
NonExpansive2 (@resize A n) := _.
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.
Program Definition list_chain
(c : chain listC) (x : A) (k : nat) : chain A :=
{| chain_car n := default x (c n !! k) |}.
Next Obligation. intros c x k n i ?. by rewrite /= (chain_cauchy c n i). Qed.
Definition list_compl `{Cofe A} : Compl listC := λ c,
match c 0 with
| [] => []
| x :: _ => compl ∘ list_chain c x <$> seq 0 (length (c 0))
end.
Global Program Instance list_cofe `{Cofe A} : Cofe listC :=
{| compl := list_compl |}.
Next Obligation.
intros ? n c; rewrite /compl /list_compl.
destruct (c 0) as [|x l] eqn:Hc0 at 1. { by destruct (chain_cauchy c 0 n); auto with lia. }
rewrite -(λ H, length_ne _ _ _ (chain_cauchy c 0 n H)); last lia.
apply Forall2_lookup=> i. rewrite -dist_option_Forall2 list_lookup_fmap.
destruct (decide (i < length (c n))); last first.
{ rewrite lookup_seq_ge ?lookup_ge_None_2; auto with lia. }
rewrite lookup_seq //= (conv_compl n (list_chain c _ _)) /=.
destruct (lookup_lt_is_Some_2 (c n) i) as [? Hcn]; first done.
by rewrite Hcn.
Qed.
Global Instance list_ofe_discrete : OfeDiscrete A → OfeDiscrete listC.
Proof. induction 2; constructor; try apply (discrete _); auto. Qed.
Global Instance nil_discrete : Discrete (@nil A).
Proof. inversion_clear 1; constructor. Qed.
Global Instance cons_discrete x l : Discrete x → Discrete l → Discrete (x :: l).
Proof. intros ??; inversion_clear 1; constructor; by apply discrete. Qed.
End cofe.
Arguments listC : clear implicits.
(** Functor *)
Lemma list_fmap_ext_ne {A} {B : ofeT} (f g : A → B) (l : list A) n :
(∀ x, f x ≡{n}≡ g x) → f <$> l ≡{n}≡ g <$> l.
Proof. intros Hf. by apply Forall2_fmap, Forall_Forall2, Forall_true. Qed.
Instance list_fmap_ne {A B : ofeT} (f : A → B) n:
Proper (dist n ==> dist n) f → Proper (dist n ==> dist n) (fmap (M:=list) f).
Proof. intros Hf l k ?; by eapply Forall2_fmap, Forall2_impl; eauto. Qed.
Definition listC_map {A B} (f : A -n> B) : listC A -n> listC B :=
CofeMor (fmap f : listC A → listC B).
Instance listC_map_ne A B : NonExpansive (@listC_map A B).
Proof. intros n f g ? l. by apply list_fmap_ext_ne. Qed.
Program Definition listCF (F : cFunctor) : cFunctor := {|
cFunctor_car A B := listC (cFunctor_car F A B);
cFunctor_map A1 A2 B1 B2 fg := listC_map (cFunctor_map F fg)
|}.
Next Obligation.
by intros F A1 A2 B1 B2 n f g Hfg; apply listC_map_ne, cFunctor_ne.
Qed.
Next Obligation.
intros F A B x. rewrite /= -{2}(list_fmap_id x).
apply list_fmap_equiv_ext=>y. apply cFunctor_id.
Qed.
Next Obligation.
intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -list_fmap_compose.
apply list_fmap_equiv_ext=>y; apply cFunctor_compose.
Qed.
Instance listCF_contractive F :
cFunctorContractive F → cFunctorContractive (listCF F).
Proof.
by intros ? A1 A2 B1 B2 n f g Hfg; apply listC_map_ne, cFunctor_contractive.
Qed.
(* CMRA *)
Section cmra.
Context {A : ucmraT}.
Implicit Types l : list A.
Local Arguments op _ _ !_ !_ / : simpl nomatch.
Instance list_op : Op (list A) :=
fix go l1 l2 := let _ : Op _ := @go in
match l1, l2 with
| [], _ => l2
| _, [] => l1
| x :: l1, y :: l2 => x ⋅ y :: l1 ⋅ l2
end.
Instance list_pcore : PCore (list A) := λ l, Some (core <$> l).
Instance list_valid : Valid (list A) := Forall (λ x, ✓ x).
Instance list_validN : ValidN (list A) := λ n, Forall (λ x, ✓{n} x).
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.
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).
Proof.
rewrite /core /= list_lookup_fmap.
destruct (l !! i); by rewrite /= ?Some_core.
Qed.
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.
Definition list_cmra_mixin : CmraMixin (list A).
Proof.
apply cmra_total_mixin.
- eauto.
- intros n l l1 l2; rewrite !list_dist_lookup=> Hl i.
by rewrite !list_lookup_op Hl.
- intros n l1 l2 Hl; by rewrite /core /= Hl.
- 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. rewrite !list_lookup_core. by apply cmra_core_mono.
- intros n l1 l2. rewrite !list_lookup_validN.
setoid_rewrite list_lookup_op. eauto using cmra_validN_op_l.
- intros n l.
induction l as [|x l IH]=> -[|y1 l1] [|y2 l2] Hl Heq;
(try by exfalso; inversion Heq).
+ by exists [], [].
+ 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..|].
exists (y1' :: l1'), (y2' :: l2'); repeat constructor; auto.
Qed.
Canonical Structure listR := CmraT (list A) list_cmra_mixin.
Global Instance list_unit : Unit (list A) := [].
Definition list_ucmra_mixin : UcmraMixin (list A).
Proof.
split.
- constructor.
- by intros l.
- by constructor.
Qed.
Canonical Structure listUR := UcmraT (list A) list_ucmra_mixin.
Global Instance list_cmra_discrete : CmraDiscrete A → CmraDiscrete listR.
Proof.
split; [apply _|]=> l; rewrite list_lookup_valid list_lookup_validN=> Hl i.
by apply cmra_discrete_valid.
Qed.
Global Instance list_core_id l : (∀ x : A, CoreId x) → CoreId l.
Proof.
intros ?; constructor; apply list_equiv_lookup=> i.
by rewrite list_lookup_core (core_id_core (l !! i)).
Qed.
(** Internalized properties *)
Lemma list_equivI {M} l1 l2 : l1 ≡ l2 ⊣⊢ (∀ i, l1 !! i ≡ l2 !! i : uPred M).
Proof. uPred.unseal; constructor=> n x ?. apply list_dist_lookup. Qed.
Lemma list_validI {M} l : ✓ l ⊣⊢ (∀ i, ✓ (l !! i) : uPred M).
Proof. uPred.unseal; constructor=> n x ?. apply list_lookup_validN. Qed.
End cmra.
Arguments listR : clear implicits.
Arguments listUR : clear implicits.
Instance list_singletonM {A : ucmraT} : SingletonM nat A (list A) := λ n x,
replicate n ε ++ [x].
Section properties.
Context {A : ucmraT}.
Implicit Types l : list A.
Implicit Types x y z : A.
Local Arguments op _ _ !_ !_ / : simpl nomatch.
Local Arguments cmra_op _ !_ !_ / : simpl nomatch.
Local Arguments ucmra_op _ !_ !_ / : simpl nomatch.
Lemma list_lookup_opM l mk i : (l ⋅? mk) !! i = l !! i ⋅ (mk ≫= (!! i)).
Proof. destruct mk; by rewrite /= ?list_lookup_op ?right_id_L. Qed.
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.
Lemma list_op_app l1 l2 l3 :
(l1 ++ l3) ⋅ l2 = (l1 ⋅ take (length l1) l2) ++ (l3 ⋅ drop (length l1) l2).
Proof. revert l2 l3.
induction l1 as [|x1 l1]=> -[|x2 l2] [|x3 l3]; f_equal/=; auto.
Qed.
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.
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.
Global Instance list_singletonM_ne i :
NonExpansive (@list_singletonM A i).
Proof. intros n l1 l2 ?. apply Forall2_app; by repeat constructor. Qed.
Global Instance list_singletonM_proper i :
Proper ((≡) ==> (≡)) (list_singletonM i) := ne_proper _.
Lemma elem_of_list_singletonM i z x : z ∈ ({[i := x]} : list A) → z = ε ∨ z = x.
Proof.
rewrite elem_of_app elem_of_list_singleton elem_of_replicate. naive_solver.
Qed.
Lemma list_lookup_singletonM i x : ({[ i := x ]} : list A) !! i = Some x.
Proof. induction i; by f_equal/=. Qed.
Lemma list_lookup_singletonM_ne i j x :
i ≠ j →
({[ i := x ]} : list A) !! j = None ∨ ({[ i := x ]} : list A) !! j = Some ε.
Proof. revert j; induction i; intros [|j]; naive_solver auto with lia. Qed.
Lemma list_singletonM_validN n i x : ✓{n} ({[ i := x ]} : list A) ↔ ✓{n} x.
Proof.
rewrite list_lookup_validN. split.
{ move=> /(_ i). by rewrite list_lookup_singletonM. }
intros Hx j; destruct (decide (i = j)); subst.
- by rewrite list_lookup_singletonM.
- destruct (list_lookup_singletonM_ne i j x) as [Hi|Hi]; first done;
rewrite Hi; by try apply (ucmra_unit_validN (A:=A)).
Qed.
Lemma list_singleton_valid i x : ✓ ({[ i := x ]} : list A) ↔ ✓ x.
Proof.
rewrite !cmra_valid_validN. by setoid_rewrite list_singletonM_validN.
Qed.
Lemma list_singletonM_length i x : length {[ i := x ]} = S i.
Proof.
rewrite /singletonM /list_singletonM app_length replicate_length /=; lia.
Qed.
Lemma list_core_singletonM i (x : A) : core {[ i := x ]} ≡ {[ i := core x ]}.
Proof.
rewrite /singletonM /list_singletonM.
by rewrite {1}/core /= fmap_app fmap_replicate (core_id_core ∅).
Qed.
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.
Lemma list_alter_singletonM f i x :
alter f i ({[i := x]} : list A) = {[i := f x]}.
Proof.
rewrite /singletonM /list_singletonM /=. induction i; f_equal/=; auto.
Qed.
Global Instance list_singleton_core_id i (x : A) :
CoreId x → CoreId {[ i := x ]}.
Proof. by rewrite !core_id_total list_core_singletonM=> ->. Qed.
(* Update *)
Lemma list_singleton_updateP (P : A → Prop) (Q : list A → Prop) x :
x ~~>: P → (∀ y, P y → Q [y]) → [x] ~~>: Q.
Proof.
rewrite !cmra_total_updateP=> Hup HQ n lf /list_lookup_validN Hv.
destruct (Hup n (default ε (lf !! 0))) as (y&?&Hv').
{ move: (Hv 0). by destruct lf; rewrite /= ?right_id. }
exists [y]; split; first by auto.
apply list_lookup_validN=> i.
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.
Qed.
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.
Lemma list_middle_update l1 l2 x y : x ~~> y → l1 ++ x :: l2 ~~> l1 ++ y :: l2.
Proof.
rewrite !cmra_update_updateP=> ?; eauto using list_middle_updateP with subst.
Qed.
(* FIXME
Lemma list_middle_local_update l1 l2 x y ml :
x ~l~> y @ ml ≫= (!! length l1) →
l1 ++ x :: l2 ~l~> l1 ++ y :: l2 @ ml.
Proof.
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.
Qed.
Lemma list_singleton_local_update i x y ml :
x ~l~> y @ ml ≫= (!! i) → {[ i := x ]} ~l~> {[ i := y ]} @ ml.
Proof. intros; apply list_middle_local_update. by rewrite replicate_length. Qed.
*)
End properties.
(** Functor *)
Instance list_fmap_cmra_morphism {A B : ucmraT} (f : A → B)
`{!CmraMorphism f} : CmraMorphism (fmap f : list A → list B).
Proof.
split; try apply _.
- intros n l. rewrite !list_lookup_validN=> Hl i. rewrite list_lookup_fmap.
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.
Qed.
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)
|}.
Next Obligation.
by intros F ???? n f g Hfg; apply listC_map_ne, urFunctor_ne.
Qed.
Next Obligation.
intros F A B x. rewrite /= -{2}(list_fmap_id x).
apply list_fmap_equiv_ext=>y. apply urFunctor_id.
Qed.
Next Obligation.
intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -list_fmap_compose.
apply list_fmap_equiv_ext=>y; apply urFunctor_compose.
Qed.
Instance listURF_contractive F :
urFunctorContractive F → urFunctorContractive (listURF F).
Proof.
by intros ? A1 A2 B1 B2 n f g Hfg; apply listC_map_ne, urFunctor_contractive.
Qed.