From iris.prelude Require Import tactics.
Set Default Proof Using "Type".
Local Set Universe Polymorphism.
(* Not using [list Type] in order to avoid universe inconsistencies *)
Inductive tlist := tnil : tlist | tcons : Type → tlist → tlist.
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 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 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, hlam (λ x, hcompose f (g x))
end.