More hlist stuff.

theories/hlist.v

-From stdpp Require Import base.
+From stdpp Require Import tactics.
 
 (* Not using [list Type] in order to avoid universe inconsistencies *)
 Inductive tlist := tnil : tlist | tcons : Type → tlist → tlist.
@@ -7,22 +7,53 @@ Inductive hlist : tlist → Type :=
   | hnil : hlist tnil
   | hcons {A As} : A → hlist As → hlist (tcons A As).
 
+Fixpoint tapp (As Bs : tlist) : tlist :=
+  match As with tnil => Bs | tcons A As => tcons A (tapp As Bs) end.
+Fixpoint happ {As Bs} (xs : hlist As) (ys : hlist Bs) : hlist (tapp As Bs) :=
+  match xs with hnil => ys | hcons _ _ x xs => hcons x (happ xs ys) end.
+
+Fixpoint hhead {A As} (xs : hlist (tcons A As)) : A :=
+  match xs with hnil => () | hcons _ _ x _ => x end.
+Fixpoint htail {A As} (xs : hlist (tcons A As)) : hlist As :=
+  match xs with hnil => () | hcons _ _ _ xs => xs end.
+
+Fixpoint hheads {As Bs} : hlist (tapp As Bs) → hlist As :=
+  match As with
+  | tnil => λ _, hnil
+  | tcons A As => λ xs, hcons (hhead xs) (hheads (htail xs))
+  end.
+Fixpoint htails {As Bs} : hlist (tapp As Bs) → hlist Bs :=
+  match As with
+  | tnil => id
+  | tcons A As => λ xs, htails (htail xs)
+  end.
+
 Fixpoint himpl (As : tlist) (B : Type) : Type :=
   match As with tnil => B | tcons A As => A → himpl As B end.
 
-Definition happly {As B} (f : himpl As B) (xs : hlist As) : B :=
+Definition hinit {B} (y : B) : himpl tnil B := y.
+Definition hlam {A As B} (f : A → himpl As B) : himpl (tcons A As) B := f.
+Arguments hlam _ _ _ _ _/.
+
+Definition hcurry {As B} (f : himpl As B) (xs : hlist As) : B :=
   (fix go As xs :=
     match xs in hlist As return himpl As B → B with
     | hnil => λ f, f
     | hcons A As x xs => λ f, go As xs (f x)
     end) _ xs f.
-Coercion happly : himpl >-> Funclass.
+Coercion hcurry : himpl >-> Funclass.
+
+Fixpoint huncurry {As B} : (hlist As → B) → himpl As B :=
+  match As with
+  | tnil => λ f, f hnil
+  | tcons x xs => λ f, hlam (λ x, huncurry (f ∘ hcons x))
+  end.
+
+Lemma hcurry_uncurry {As B} (f : hlist As → B) xs : huncurry f xs = f xs.
+Proof. by induction xs as [|A As x xs IH]; simpl; rewrite ?IH. Qed.
 
 Fixpoint hcompose {As B C} (f : B → C) {struct As} : himpl As B → himpl As C :=
   match As with
   | tnil => f
-  | tcons A As => λ g x, hcompose f (g x)
+  | tcons A As => λ g, hlam (λ x, hcompose f (g x))
   end.
-
-Definition hinit {B} (y : B) : himpl tnil B := y.
-Definition hlam {A As B} (f : A → himpl As B) : himpl (tcons A As) B := f.