Set Implicit Arguments.
| Streams |
Section Streams.
Variable A : Set.
CoInductive Stream : Set :=
Cons : A -> Stream -> Stream.
Definition hd (x:Stream) := match x with
| Cons a _ => a
end.
Definition tl (x:Stream) := match x with
| Cons _ s => s
end.
Fixpoint Str_nth_tl (n:nat) (s:Stream) {struct n} : Stream :=
match n with
| O => s
| S m => Str_nth_tl m (tl s)
end.
Definition Str_nth (n:nat) (s:Stream) : A := hd (Str_nth_tl n s).
Lemma unfold_Stream :
forall x:Stream, x = match x with
| Cons a s => Cons a s
end.
Proof.
intro x.
case x.
trivial.
Qed.
Lemma tl_nth_tl :
forall (n:nat) (s:Stream), tl (Str_nth_tl n s) = Str_nth_tl n (tl s).
Proof.
simple induction n; simpl in |- *; auto.
Qed.
Hint Resolve tl_nth_tl: datatypes v62.
Lemma Str_nth_tl_plus :
forall (n m:nat) (s:Stream),
Str_nth_tl n (Str_nth_tl m s) = Str_nth_tl (n + m) s.
simple induction n; simpl in |- *; intros; auto with datatypes.
rewrite <- H.
rewrite tl_nth_tl; trivial with datatypes.
Qed.
Lemma Str_nth_plus :
forall (n m:nat) (s:Stream), Str_nth n (Str_nth_tl m s) = Str_nth (n + m) s.
intros; unfold Str_nth in |- *; rewrite Str_nth_tl_plus;
trivial with datatypes.
Qed.
| Extensional Equality between two streams |
CoInductive EqSt : Stream -> Stream -> Prop :=
eqst :
forall s1 s2:Stream,
hd s1 = hd s2 -> EqSt (tl s1) (tl s2) -> EqSt s1 s2.
| A coinduction principle |
Ltac coinduction proof :=
cofix proof; intros; constructor;
[ clear proof | try (apply proof; clear proof) ].
| Extensional equality is an equivalence relation |
Theorem EqSt_reflex : forall s:Stream, EqSt s s.
coinduction EqSt_reflex.
reflexivity.
Qed.
Theorem sym_EqSt : forall s1 s2:Stream, EqSt s1 s2 -> EqSt s2 s1.
coinduction Eq_sym.
case H; intros; symmetry in |- *; assumption.
case H; intros; assumption.
Qed.
Theorem trans_EqSt :
forall s1 s2 s3:Stream, EqSt s1 s2 -> EqSt s2 s3 -> EqSt s1 s3.
coinduction Eq_trans.
transitivity (hd s2).
case H; intros; assumption.
case H0; intros; assumption.
apply (Eq_trans (tl s1) (tl s2) (tl s3)).
case H; trivial with datatypes.
case H0; trivial with datatypes.
Qed.
| The definition given is equivalent to require the elements at each position to be equal |
Theorem eqst_ntheq :
forall (n:nat) (s1 s2:Stream), EqSt s1 s2 -> Str_nth n s1 = Str_nth n s2.
unfold Str_nth in |- *; simple induction n.
intros s1 s2 H; case H; trivial with datatypes.
intros m hypind.
simpl in |- *.
intros s1 s2 H.
apply hypind.
case H; trivial with datatypes.
Qed.
Theorem ntheq_eqst :
forall s1 s2:Stream,
(forall n:nat, Str_nth n s1 = Str_nth n s2) -> EqSt s1 s2.
coinduction Equiv2.
apply (H 0).
intros n; apply (H (S n)).
Qed.
Section Stream_Properties.
Variable P : Stream -> Prop.
Inductive Exists : Stream -> Prop :=
| Here : forall x:Stream, P x -> Exists x
| Further : forall x:Stream, Exists (tl x) -> Exists x.
CoInductive ForAll : Stream -> Prop :=
HereAndFurther : forall x:Stream, P x -> ForAll (tl x) -> ForAll x.
Section Co_Induction_ForAll.
Variable Inv : Stream -> Prop.
Hypothesis InvThenP : forall x:Stream, Inv x -> P x.
Hypothesis InvIsStable : forall x:Stream, Inv x -> Inv (tl x).
Theorem ForAll_coind : forall x:Stream, Inv x -> ForAll x.
coinduction ForAll_coind; auto.
Qed.
End Co_Induction_ForAll.
End Stream_Properties.
End Streams.
Section Map.
Variables A B : Set.
Variable f : A -> B.
CoFixpoint map (s:Stream A) : Stream B := Cons (f (hd s)) (map (tl s)).
End Map.
Section Constant_Stream.
Variable A : Set.
Variable a : A.
CoFixpoint const : Stream A := Cons a const.
End Constant_Stream.
Unset Implicit Arguments.