Remove old stuff.

theories/logrel/kind_tele.v

@@ -6,14 +6,14 @@ Set Default Proof Using "Type".
 dependencies between binders, hence we might have just used a list of [kind]
 but might be needed for future extensions, such as for bounded polymorphism *)
 (** Type Telescopes *)
-Inductive ktele {Σ} : Type :=
+Inductive ktele {Σ} :=
   | KTeleO : ktele
   | KTeleS {k} (binder : lty Σ k → ktele) : ktele.
 
 Arguments ktele : clear implicits.
 
 (** The telescope version of kind types *)
-Fixpoint ktele_fun {Σ} (kt : ktele Σ) (T : Type) : Type :=
+Fixpoint ktele_fun {Σ} (kt : ktele Σ) (T : Type) :=
   match kt with
   | KTeleO => T
   | KTeleS b => ∀ K, ktele_fun (b K) T
@@ -22,9 +22,7 @@ Fixpoint ktele_fun {Σ} (kt : ktele Σ) (T : Type) : Type :=
 Notation "kt -k> A" :=
   (ktele_fun kt A) (at level 99, A at level 200, right associativity).
 
-(** An eliminator for elements of [ktele_fun].
-    We use a [fix] because, for some reason, that makes stuff print nicer
-    in the proofs in iris:bi/lib/telescopes.v *)
+(** An eliminator for elements of [ktele_fun]. *)
 Definition ktele_fold {Σ X Y kt}
     (step : ∀ {k}, (lty Σ k → Y) → Y) (base : X → Y) : (kt -k> X) → Y :=
   (fix rec {kt} : (kt -k> X) → Y :=
@@ -52,10 +50,6 @@ Definition ktele_app {Σ kt T} (f : kt -k> T) : ltys Σ kt → T :=
      end) kt Ks f.
 Arguments ktele_app {_} {!_ _} _ !_ /.
 
-(* Coercion ltys : ktele >-> Sortclass. *)
-(* This is a local coercion because otherwise, the "λ.." notation stops working. *)
-(* Local Coercion ktele_app : ktele_fun >-> Funclass. *)
-
 (** Inversion lemma for [tele_arg] *)
 Lemma ltys_inv {Σ kt} (Ks : ltys Σ kt) :
   match kt as kt return ltys _ kt → Prop with
@@ -68,31 +62,7 @@ Proof. exact (ltys_inv Ks). Qed.
 Lemma ltys_S_inv {Σ X} {f : lty Σ X → ktele Σ} (Ks : ltys Σ (KTeleS f)) :
   ∃ K Ks', Ks = LTysS K Ks'.
 Proof. exact (ltys_inv Ks). Qed.
-(*
-(** Map below a tele_fun *)
-Fixpoint ktele_map {Σ} {T U} {kt : ktele Σ} : (T → U) → (kt -k> T) → kt -k> U :=
-  match kt as kt return (T → U) → (kt -k> T) → kt -k> U with
-  | KTeleO => λ F : T → U, F
-  | @KTeleS _ X b => λ (F : T → U) (f : KTeleS b -k> T) (x : lty Σ X),
-                  ktele_map F (f x)
-  end.
-Arguments ktele_map {_} {_ _ !_} _ _ /.
-Lemma ktele_map_app {Σ} {T U} {kt : ktele Σ} (F : T → U) (t : kt -k> T) (x : kt) :
-  (ktele_map F t) x = F (t x).
-Proof.
-  induction kt as [|X f IH]; simpl in *.
-  - rewrite (ltys_O_inv x). done.
-  - destruct (ltys_S_inv x) as [x' [a' ->]]. simpl.
-    rewrite <-IH. done.
-Qed.
-
-Global Instance ktele_fmap {Σ} {kt : ktele Σ} : FMap (ktele_fun kt) :=
-  λ T U, ktele_map.
 
-Lemma ktele_fmap_app {Σ} {T U} {kt : ktele Σ} (F : T → U) (t : kt -k> T) (x : kt) :
-  (F <$> t) x = F (t x).
-Proof. apply ktele_map_app. Qed.
-*)
 (** Operate below [tele_fun]s with argument telescope [kt]. *)
 Fixpoint ktele_bind {Σ U kt} : (ltys Σ kt → U) → kt -k> U :=
   match kt as kt return (ltys _ kt → U) → kt -k> U with
@@ -111,74 +81,3 @@ Proof.
   - destruct (ltys_S_inv K) as [K' [Ks' ->]]. simpl.
     rewrite IH. done.
 Qed.
-
-Fixpoint ktele_to_tele {Σ} (kt : ktele Σ) : tele :=
-  match kt with
-  | KTeleO => TeleO
-  | KTeleS b => TeleS (λ x, ktele_to_tele (b x))
-  end.
-
-Fixpoint ltys_to_tele_args {Σ} {kt} (Ks : ltys Σ kt) :
-    tele_arg (ktele_to_tele kt) :=
-  match Ks with
-  | LTysO => TargO
-  | LTysS K Ks => TargS K (ltys_to_tele_args Ks)
-  end.
-
-(*
-
-(** We can define the identity function and composition of the [-t>] function *)
-(* space. *)
-Definition ktele_fun_id {Σ} {kt : ktele Σ} : kt -k> kt := ktele_bind id.
-
-Lemma ktele_fun_id_eq {Σ} {kt : ktele Σ} (x : kt) :
-  ktele_fun_id x = x.
-Proof. unfold ktele_fun_id. rewrite ktele_app_bind. done. Qed.
-
-Definition ktele_fun_compose {Σ} {kt1 kt2 kt3 : ktele Σ} :
-  (kt2 -k> kt3) → (kt1 -k> kt2) → (kt1 -k> kt3) :=
-  λ t1 t2, ktele_bind (compose (ktele_app t1) (ktele_app t2)).
-
-Lemma ktele_fun_compose_eq {Σ} {kt1 kt2 kt3 : ktele Σ} (f : kt2 -k> kt3) (g : kt1 -k> kt2) x :
-  ktele_fun_compose f g $ x = (f ∘ g) x.
-Proof. unfold ktele_fun_compose. rewrite ktele_app_bind. done. Qed.
-*)
-
-(*
-(** Notation-compatible telescope mapping *)
-(* This adds (tele_app ∘ tele_bind), which is an identity function, around every *)
-(*    binder so that, after simplifying, this matches the way we typically write *)
-(*    notations involving telescopes. *)
-Notation "'λ..' x .. y , e" :=
-  (ktele_app (ktele_bind (λ x, .. (ktele_app (ktele_bind (λ y, e))) .. )))
-  (at level 200, x binder, y binder, right associativity,
-   format "'[  ' 'λ..'  x  ..  y ']' ,  e") : stdpp_scope.
-
-
-(** Telescopic quantifiers *)
-Definition ktforall {Σ} {kt : ktele Σ} (Ψ : kt → Prop) : Prop :=
-  ktele_fold (λ (T : kind), λ (b : (lty Σ T) → Prop), (∀ x : (lty Σ T), b x)) (λ x, x) (ktele_bind Ψ).
-Arguments ktforall {_ !_} _ /.
-
-Notation "'∀..' x .. y , P" := (ktforall (λ x, .. (ktforall (λ y, P)) .. ))
-  (at level 200, x binder, y binder, right associativity,
-  format "∀..  x  ..  y ,  P") : stdpp_scope.
-
-Lemma ktforall_forall {Σ} {kt : ktele Σ} (Ψ : kt → Prop) :
-  ktforall Ψ ↔ (∀ x, Ψ x).
-Proof.
-  symmetry. unfold ktforall. induction kt as [|X ft IH].
-  - simpl. split.
-    + done.
-    + intros ? p. rewrite (ltys_O_inv p). done.
-  - simpl. split; intros Hx a.
-    + rewrite <-IH. done.
-    + destruct (ltys_S_inv a) as [x [pf ->]].
-      revert pf. setoid_rewrite IH. done.
-Qed.
-
-(* Teach typeclass resolution how to make progress on these binders *)
-Typeclasses Opaque ktforall.
-Hint Extern 1 (ktforall _) =>
-  progress cbn [ttforall ktele_fold ktele_bind ktele_app] : typeclass_instances.
-*)