Commit 3c9e6825 by Robbert Krebbers

### A more understandable proof that lists form a COFE.

```This proof also more easily scales to other recursive types, like
trees etc.```
parent 17bc1af5
 ... ... @@ -11,5 +11,5 @@ install: [make "install"] remove: ["rm" "-rf" "%{lib}%/coq/user-contrib/iris"] depends: [ "coq" { (>= "8.7.1" & < "8.10~") | (= "dev") } "coq-stdpp" { (= "1.2.0") | (= "dev") } "coq-stdpp" { (= "dev.2019-05-03.0.09f69b7c") | (= "dev") } ]
 ... ... @@ -19,7 +19,7 @@ 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_head_ne : NonExpansive (head (A:=A)). Global Instance head_ne : NonExpansive (head (A:=A)). Proof. destruct 1; by constructor. Qed. Global Instance list_lookup_ne i : NonExpansive (lookup (M:=list A) i). ... ... @@ -63,28 +63,30 @@ Proof. 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 (** To define [compl : chain (list A) → list A] we make use of the fact that given a given chain [c0, c1, c2, ...] of lists, the list [c0] completely determines the shape (i.e. the length) of all lists in the chain. So, the [compl] operation is defined by structural recursion on [c0], and takes the completion of the elements of all lists in the chain point-wise. We use [head] and [tail] as the inverse of [cons]. *) Fixpoint list_compl_go `{!Cofe A} (c0 : list A) (c : chain listC) : listC := match c0 with | [] => [] | x :: _ => compl ∘ list_chain c x <\$> seq 0 (length (c 0)) | x :: c0 => compl (chain_map (default x ∘ head) c) :: list_compl_go c0 (chain_map tail c) end. Global Program Instance list_cofe `{!Cofe A} : Cofe listC := {| compl := list_compl |}. {| compl c := list_compl_go (c 0) c |}. 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. intros ? n c; rewrite /compl. assert (c 0 ≡{0}≡ c n) as Hc0 by (symmetry; apply chain_cauchy; lia). revert Hc0. generalize (c 0)=> c0. revert c. induction c0 as [|x c0 IH]=> c Hc0 /=. { by inversion Hc0. } apply list_dist_cons_inv_l in Hc0 as (x' & xs' & Hx & Hc0 & Hcn). rewrite Hcn. f_equiv. - by rewrite conv_compl /= Hcn /=. - rewrite IH /= ?Hcn //. Qed. Global Instance list_ofe_discrete : OfeDiscrete A → OfeDiscrete listC. ... ...
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