Revert "define Earlier: kind of the 'opposite' of later"

This reverts commit 327acdbd. I thought we could
get something with a contractive "un-injection" funtion, but that turns out to
be impossible.

algebra/cofe.v

@@ -464,14 +464,13 @@ Section later.
     match n with 0 => True | S n => later_car x ≡{n}≡ later_car y end.
   Program Definition later_chain (c : chain (later A)) : chain A :=
     {| chain_car n := later_car (c (S n)) |}.
-  Next Obligation. intros c n i ?. apply (chain_cauchy c (S n)); lia. Qed.
+  Next Obligation. intros c n i ?; apply (chain_cauchy c (S n)); lia. Qed.
   Instance later_compl : Compl (later A) := λ c, Next (compl (later_chain c)).
   Definition later_cofe_mixin : CofeMixin (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)).
+    - intros x y; unfold equiv, later_equiv; rewrite !equiv_dist.
+      split. intros Hxy [|n]; [done|apply Hxy]. intros Hxy n; apply (Hxy (S n)).
     - intros [|n]; [by split|split]; unfold dist, later_dist.
       + by intros [x].
       + by intros [x] [y].
@@ -530,48 +529,6 @@ Proof.
   apply laterC_map_contractive => i ?. by apply cFunctor_ne, Hfg.
 Qed.
 
-(** Earlier *)
-Inductive earlier (A : Type) : Type := Prev { earlier_car : A }.
-Add Printing Constructor earlier.
-Arguments Prev {_} _.
-Arguments earlier_car {_} _.
-Lemma earlier_eta {A} (x : earlier A) : Prev (earlier_car x) = x.
-Proof. by destruct x. Qed.
-
-Section earlier.
-  Context {A : cofeT}.
-  Instance earlier_equiv : Equiv (earlier A) :=
-    λ x y, earlier_car x ≡ earlier_car y.
-  Instance earlier_dist : Dist (earlier A) :=
-    λ n x y, earlier_car x ≡{S n}≡ earlier_car y.
-  Program Definition earlier_chain (c : chain (earlier A)) : chain A :=
-    {| chain_car n := earlier_car (c n) |}.
-  Next Obligation. intros c n i ?. by apply dist_S, (chain_cauchy c). Qed.
-  Instance earlier_compl : Compl (earlier A) :=
-    λ c, Prev (compl (earlier_chain c)).
-  Definition earlier_cofe_mixin : CofeMixin (earlier A).
-  Proof.
-    split.
-    - intros x y; unfold equiv, earlier_equiv; rewrite !equiv_dist. split.
-      + intros Hxy n; apply dist_S, Hxy.
-      + intros Hxy [|n]; [apply dist_S, Hxy|apply Hxy].
-    - intros n; split; unfold dist, earlier_dist.
-      + by intros [x].
-      + by intros [x] [y].
-      + by intros [x] [y] [z] ??; trans y.
-    - intros n [x] [y] ?; unfold dist, earlier_dist; by apply dist_S.
-    - intros n c. rewrite /compl /earlier_compl /dist /earlier_dist /=.
-      rewrite (conv_compl (S n) (earlier_chain c)). simpl.
-      apply (chain_cauchy c n). omega.
-  Qed.
-  Canonical Structure earlierC : cofeT := CofeT earlier_cofe_mixin.
-  Global Instance Earlier_inj n : Inj (dist (S n)) (dist n) (@Prev A).
-  Proof. by intros x y. Qed.
-End earlier.
-
-Arguments earlierC : clear implicits.
-
-
 (** Notation for writing functors *)
 Notation "∙" := idCF : cFunctor_scope.
 Notation "F1 -n> F2" := (cofe_morCF F1%CF F2%CF) : cFunctor_scope.