theories/Lists

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+
+(* $Id$ *)
+
+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.