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Require Import Program.Tactics.
Require Import FunctionalExtensionality.
Inductive Succ A :=
| Succ_O : Succ A
| Succ_S : A -> Succ A.
Arguments Succ_O {A}.
Arguments Succ_S {A} _.
Inductive Zero : Type :=.
Inductive ty :=
| ty_nat : ty
| ty_arr : ty -> ty -> ty.
Inductive term A :=
| term_abs : ty -> term (Succ A) -> term A
| term_app : term A -> term A -> term A
| term_var : A -> term A
| term_nat : nat -> term A.
Arguments term_abs {A} _ _.
Arguments term_app {A} _ _.
Arguments term_var {A} _.
Arguments term_nat {A} _.
Class Functor F :=
{
ret : forall {A}, A -> F A;
fmap : forall {A B}, (A -> B) -> F A -> F B;
fmap_id : forall {A} (fa : F A), fmap (@id A) fa = fa;
fmap_compose : forall {A B C} (fa : F A) (g : B -> C) (h : A -> B), fmap (fun
a => g (h a)) fa = fmap g (fmap h fa)
}.
Class Monad M `{Functor M} :=
{
bind : forall {A B}, M A -> (A -> M B) -> M B;
ret_left_id : forall {A B} (a : A) (f : A -> M B), bind (ret a) f = f a;
ret_right_id : forall {A} (m : M A), bind m ret = m;
bind_assoc : forall {A B C} (m : M A) (f : A -> M B) (g : B -> M C), bind
(bind m f) g = bind m (fun x => bind (f x) g)
}.
Instance Succ_Functor : Functor Succ.
Proof.
unshelve econstructor.
- intros A B f fa.
destruct fa.
+ apply Succ_O.
+ apply Succ_S. tauto.
- intros. apply Succ_S. assumption.
- intros A [|a]; reflexivity.
- intros A B C [|a] g h; reflexivity.
Defined.
(* Anomaly: Not an arity *)
Program Fixpoint term_bind {A B} (tm : term A) (f : A -> term B) : term B :=
let Succ_f (var : Succ A) :=
match var with
| Succ_O => term_var Succ_O
| Succ_S var' => _
end in
match tm with
| term_app tm1 tm2 => term_app (term_bind tm1 f) (term_bind tm2 f)
| term_abs ty body => term_abs ty (term_bind body Succ_f)
| term_var a => f a
| term_nat n => term_nat n
end.
Next Obligation.
intros.
Admitted.
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