add telescopes and a bit of theory about them

_CoqProject

@@ -42,4 +42,4 @@ theories/sorting.v
 theories/infinite.v
 theories/nat_cancel.v
 theories/namespaces.v
-
+theories/telescopes.v

theories/telescopes.v

+From stdpp Require Import base tactics.
+Set Default Proof Using "Type".
+
+(** Telescopes *)
+Inductive tele : Type :=
+  | TeleO : tele
+  | TeleS {X} (binder : X → tele) : tele.
+
+Arguments TeleS {_} _.
+
+(** The telescope version of Coq's function type *)
+Fixpoint tele_fun (TT : tele) (T : Type) : Type :=
+  match TT with
+  | TeleO => T
+  | TeleS b => ∀ x, tele_fun (b x) T
+  end.
+
+Notation "TT -t> A" :=
+  (tele_fun TT A) (at level 99, A at level 200, right associativity).
+
+(** An eliminator for elements of [tele_fun].
+    We use a [fix] because, for some reason, that makes stuff print nicer
+    in the proofs in iris:bi/lib/telescopes.v *)
+Definition tele_fold {X Y} {TT : tele} (step : ∀ {A : Type}, (A → Y) → Y) (base : X → Y)
+  : (TT -t> X) → Y :=
+  (fix rec {TT} : (TT -t> X) → Y :=
+     match TT as TT return (TT -t> X) → Y with
+     | TeleO => λ x : X, base x
+     | TeleS b => λ f, step (λ x, rec (f x))
+     end) TT.
+Arguments tele_fold {_ _ !_} _ _ _ /.
+
+(** A sigma-like type for an "element" of a telescope, i.e. the data it
+  takes to get a [T] from a [TT -t> T]. *)
+Inductive tele_arg : tele → Type :=
+| TargO : tele_arg TeleO
+(* the [x] is the only relevant data here *)
+| TargS {X} {binder} (x : X) : tele_arg (binder x) → tele_arg (TeleS binder).
+
+Definition tele_app {TT : tele} {T} (f : TT -t> T) : tele_arg TT → T :=
+  λ a, (fix rec {TT} (a : tele_arg TT) : (TT -t> T) → T :=
+     match a in tele_arg TT return (TT -t> T) → T with
+     | TargO => λ t : T, t
+     | TargS x a => λ f, rec a (f x)
+     end) TT a f.
+Arguments tele_app {!_ _} _ !_ /.
+
+Coercion tele_arg : tele >-> Sortclass.
+Coercion tele_app : tele_fun >-> Funclass.
+
+(** Inversion lemma for [tele_arg] *)
+Lemma tele_arg_inv {TT : tele} (a : TT) :
+  match TT as TT return TT → Prop with
+  | TeleO => λ a, a = TargO
+  | TeleS f => λ a, ∃ x a', a = TargS x a'
+  end a.
+Proof. induction a; eauto. Qed.
+Lemma tele_arg_O_inv (a : TeleO) : a = TargO.
+Proof. exact (tele_arg_inv a). Qed.
+Lemma tele_arg_S_inv {X} {f : X → tele} (a : TeleS f) :
+  ∃ x a', a = TargS x a'.
+Proof. exact (tele_arg_inv a). Qed.
+
+(** Map below a tele_fun *)
+Fixpoint tele_map {T U} {TT : tele} : (T → U) → (TT -t> T) → TT -t> U :=
+  match TT as TT return (T → U) → (TT -t> T) → TT -t> U with
+  | TeleO => λ F : T → U, F
+  | @TeleS X b => λ (F : T → U) (f : TeleS b -t> T) (x : X),
+                  tele_map F (f x)
+  end.
+Arguments tele_map {_ _ !_} _ _ /.
+
+Lemma tele_map_app {T U} {TT : tele} (F : T → U) (t : TT -t> T) (x : TT) :
+  (tele_map F t) x = F (t x).
+Proof.
+  induction TT as [|X f IH]; simpl in *.
+  - rewrite (tele_arg_O_inv x). done.
+  - destruct (tele_arg_S_inv x) as [x' [a' ->]]. simpl.
+    rewrite <-IH. done.
+Qed.
+
+Global Instance tele_fmap {TT : tele} : FMap (tele_fun TT) := λ T U, tele_map.
+
+Lemma tele_fmap_app {T U} {TT : tele} (F : T → U) (t : TT -t> T) (x : TT) :
+  (F <$> t) x = F (t x).
+Proof. apply tele_map_app. Qed.
+
+Global Instance tele_fmap2 {TT1 TT2 : tele} : FMap (tele_fun TT1 ∘ tele_fun TT2) :=
+  λ T U, tele_map ∘ tele_map.
+
+Lemma tele_fmap2_app {T U} {TT1 TT2 : tele} (F : T → U) (t : TT1 -t> TT2 -t> T)
+      (x : TT1) (y : TT2) :
+  (F <$> t) x y = F (t x y).
+Proof.
+  unfold fmap, tele_fmap2. simpl.
+  rewrite !tele_map_app. done.
+Qed.
+
+(** Operate below [tele_fun]s with argument telescope [TT]. *)
+Fixpoint tele_bind {U} {TT : tele} : (TT → U) → TT -t> U :=
+  match TT as TT return (TT → U) → TT -t> U with
+  | TeleO => λ F, F TargO
+  | @TeleS X b => λ (F : TeleS b → U) (x : X), (* b x -t> U *)
+                  tele_bind (λ a, F (TargS x a))
+  end.
+Arguments tele_bind {_ !_} _ /.
+
+(** A function that looks funny. *)
+Definition tele_arg_id (TT : tele) : TT -t> TT := tele_bind id.
+
+(** Notation *)
+Notation "'[tele' x .. z ]" :=
+  (TeleS (fun x => .. (TeleS (fun z => TeleO)) ..))
+  (x binder, z binder, format "[tele  '[hv' x .. z ']' ]").
+Notation "'[tele' ]" := (TeleO)
+  (format "[tele ]").
+
+Notation "'[tele_arg' x ; .. ; z ]" :=
+  (TargS x ( .. (TargS z TargO) ..))
+  (format "[tele_arg  '[hv' x ;  .. ;  z ']' ]").
+Notation "'[tele_arg' ]" := (TargO)
+  (format "[tele_arg ]").
+
+(** Notation-compatible telescope mapping *)
+Notation "'λ..' x .. y , e" :=
+  (tele_app $ tele_bind (λ x, .. (tele_app $ tele_bind (λ y, e)) .. ))
+  (at level 200, x binder, y binder, right associativity,
+   format "'[  ' 'λ..'  x  ..  y ']' ,  e").