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Require Import Program.
Set Primitive Projections.
CoInductive Stream (A : Type) := mkStream { hd : A; tl : Stream A}.
Arguments mkStream [A] hd tl.
Arguments hd [A] s.
Arguments tl [A] s.
Definition eta {A} (s : Stream A) := {| hd := s.(hd); tl := s.(tl) |}.
CoFixpoint ones := {| hd := 1; tl := ones |}.
CoFixpoint ticks := {| hd := tt; tl := ticks |}.
CoInductive stream_equiv {A} (s : Stream A) (s' : Stream A) : Prop :=
mkStreamEq { hdeq : s.(hd) = s'.(hd); tleq : stream_equiv s.(tl) s'.(tl) }.
Arguments hdeq {A} {s} {s'}.
Arguments tleq {A} {s} {s'}.
Program CoFixpoint ones_eq : stream_equiv ones ones.(tl) :=
{| hdeq := eq_refl; tleq := ones_eq |}.
CoFixpoint stream_equiv_refl {A} (s : Stream A) : stream_equiv s s :=
{| hdeq := eq_refl; tleq := stream_equiv_refl (tl s) |}.
CoFixpoint stream_equiv_sym {A} (s s' : Stream A) (H : stream_equiv s s') : stream_equiv s' s :=
{| hdeq := eq_sym H.(hdeq); tleq := stream_equiv_sym _ _ H.(tleq) |}.
CoFixpoint stream_equiv_trans {A} {s s' s'' : Stream A}
(H : stream_equiv s s') (H' : stream_equiv s' s'') : stream_equiv s s'' :=
{| hdeq := eq_trans H.(hdeq) H'.(hdeq);
tleq := stream_equiv_trans H.(tleq) H'.(tleq) |}.
Program Definition eta_eq {A} (s : Stream A) : stream_equiv s (eta s):=
{| hdeq := eq_refl; tleq := stream_equiv_refl (tl (eta s))|}.
Section Parks.
Variable A : Type.
Variable R : Stream A -> Stream A -> Prop.
Hypothesis bisim1 : forall s1 s2:Stream A,
R s1 s2 -> hd s1 = hd s2.
Hypothesis bisim2 : forall s1 s2:Stream A,
R s1 s2 -> R (tl s1) (tl s2).
CoFixpoint park_ppl :
forall s1 s2:Stream A, R s1 s2 -> stream_equiv s1 s2 :=
fun s1 s2 (p : R s1 s2) =>
mkStreamEq _ _ _ (bisim1 s1 s2 p)
(park_ppl (tl s1)
(tl s2)
(bisim2 s1 s2 p)).
End Parks.
Program CoFixpoint iterate {A} (f : A -> A) (x : A) : Stream A :=
{| hd := x; tl := iterate f (f x) |}.
Program CoFixpoint map {A B} (f : A -> B) (s : Stream A) : Stream B :=
{| hd := f s.(hd); tl := map f s.(tl) |}.
Theorem map_iterate A (f : A -> A) (x : A) : stream_equiv (iterate f (f x))
(map f (iterate f x)).
Proof.
apply park_ppl with
(R:= fun s1 s2 => exists x : A, s1 = iterate f (f x) /\
s2 = map f (iterate f x)).
now intros s1 s2 (x0,(->,->)).
intros s1 s2 (x0,(->,->)).
now exists (f x0).
now exists x.
Qed.
Fail Check (fun A (s : Stream A) => eq_refl : s = eta s).
Notation convertible x y := (eq_refl x : x = y).
Fail Check convertible ticks {| hd := hd ticks; tl := tl ticks |}.
CoInductive U := inU
{ outU : U }.
CoFixpoint u : U :=
inU u.
CoFixpoint force (u : U) : U :=
inU (outU u).
Lemma eq (x : U) : x = force x.
Proof.
Fail destruct x.
Abort.
(* Impossible *)
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