### Streams.v

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(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA        LRI-CNRS        ENS-CNRS                *)
(*              Rocquencourt         Orsay          Lyon                    *)
(*                                                                          *)
(*                                 Coq V6.3                                 *)
(*                               July 1st 1999                              *)
(*                                                                          *)
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(*                                Streams.v                                 *)
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Implicit Arguments On.

Section Streams. (* The set of streams : definition *)

Variable A : Set.

CoInductive Set Stream := Cons : A->Stream->Stream.

Definition
hd :=
[x:Stream] Cases x of (Cons a _) => a end.

Definition
tl :=
[x:Stream] Cases x of (Cons _ s) => s end.

Fixpoint
Str_nth_tl [n:nat] : Stream->Stream :=
[s:Stream] Cases n of
O => s
|(S m) => (
Str_nth_tl m (tl s))
end.

Definition Str_nth : nat->Stream->A := [n:nat][s:Stream](hd (Str_nth_tl n s)).

Lemma unfold_Stream :(x:Stream)x=(Cases x of (Cons a s) => (Cons a s) end).
Proof.
Intro x.
Case x.
Trivial.
Qed.

Lemma
tl_nth_tl : (n:nat)(s:Stream)(tl (Str_nth_tl n s))=(Str_nth_tl n (tl s)).
Proof.
Induction n; Simpl; Auto.
Save.
Hints Resolve tl_nth_tl : datatypes v62.

Lemma Str_nth_tl_plus
: (n,m:nat)(s:Stream)(
Str_nth_tl n (Str_nth_tl m s))=(Str_nth_tl (plus n m) s).
Induction n; Simpl; Intros; Auto with datatypes.
Rewrite <- H.
Rewrite tl_nth_tl; Trivial with datatypes.
Save.

Lemma Str_nth_plus
: (n,m:nat)(s:Stream)(
Str_nth n (Str_nth_tl m s))=(Str_nth (plus n m) s).
Intros; Unfold Str_nth; Rewrite Str_nth_tl_plus; Trivial with datatypes.
Save.

(* Extensional Equality between two streams  *)

CoInductive EqSt : Stream->Stream->Prop :=
eqst : (s1,s2:Stream)
((
hd s1)=(hd s2))->
(EqSt (tl s1) (tl s2))
->(EqSt s1 s2).

(* A coinduction principle *)

Tactic Definition CoInduction [\$proof] :=
[ <:tactic:<( Cofix \$proof; (Intros; (Constructor;
[Clear \$proof | Try (Apply \$proof;Clear \$proof)])))>> ].

(* Extensional equality is an equivalence relation *)

Theorem
EqSt_reflex : (s:Stream)(EqSt s s).
CoInduction EqSt_reflex.
Reflexivity.
Qed.

Theorem sym_EqSt :
(s1:Stream)(s2:Stream)(
EqSt s1 s2)->(EqSt s2 s1).
CoInduction Eq_sym.
(Case H;Intros;Symmetry;Assumption).
(Case H;Intros;Assumption).
Qed.

Theorem trans_EqSt :
(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 :
(n:nat)(s1,s2:Stream)(
EqSt s1 s2)->(Str_nth n s1)=(Str_nth n s2).
Unfold Str_nth; Induction n.
Intros s1 s2 H; Case H; Trivial with datatypes.
Intros m hypind.
Simpl.
Intros s1 s2 H.
Apply hypind.
(Case H; Trivial with datatypes).
Qed.

Theorem ntheq_eqst :
(s1,s2:Stream)((n:nat)(
Str_nth n s1)=(Str_nth n s2))->(EqSt s1 s2).
CoInduction Equiv2.
Apply (H O).
Intros n; Apply (H (S n)).
Qed.

Section Stream_Properties.

Variable P : Stream->Prop.

(*
Inductive   Exists :  Stream -> Prop :=
Here    : (x:Stream)(P x) ->(Exists x) |
Further : (x:Stream)~(P x)->(Exists (tl x))->(Exists x).
*)

Inductive Exists : Stream -> Prop :=
Here : (x:Stream)(P x) ->(
Exists x) |
Further : (x:Stream)(Exists (tl x))->(Exists x).

CoInductive ForAll : Stream -> Prop :=
forall : (x:Stream)(P x)->(
ForAll (tl x))->(ForAll x).

Section Co_Induction_ForAll.
Variable Inv : Stream -> Prop.
Hypothesis InvThenP : (x:Stream)(Inv x)->(P x).
Hypothesis InvIsStable: (x:Stream)(Inv x)->(Inv (tl x)).

Theorem ForAll_coind : (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 : (Stream A)->(Stream B) :=
[s:(Stream A)](Cons (f (
hd s)) (map (tl s))).
End Map.

Section Constant_Stream.
Variables A : Set.
Variable a : A.
CoFixpoint const : (Stream A) := (Cons a const).
End Constant_Stream.

Implicit Arguments Off.

(* \$Id: Streams.html,v 1.2 2005/11/19 06:27:16 pouzet Exp \$ *)

05/07/101, 15:56