1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
|
(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Set Implicit Arguments.
(** Streams *)
Section Streams.
Variable A : Type.
CoInductive Stream : Type :=
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) : 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; auto.
Qed.
Hint Resolve tl_nth_tl: datatypes.
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; 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; rewrite Str_nth_tl_plus;
trivial with datatypes.
Qed.
(** Extensional Equality between two streams *)
CoInductive EqSt (s1 s2: Stream) : Prop :=
eqst :
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 ; 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; simple 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 :
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.
(*i
Inductive Exists : Stream -> Prop :=
| Here : forall x:Stream, P x -> Exists x
| Further : forall x:Stream, ~ P x -> Exists (tl x) -> Exists x.
i*)
Inductive Exists ( x: Stream ) : Prop :=
| Here : P x -> Exists x
| Further : Exists (tl x) -> Exists x.
CoInductive ForAll (x: Stream) : Prop :=
HereAndFurther : P x -> ForAll (tl x) -> ForAll x.
Lemma ForAll_Str_nth_tl : forall m x, ForAll x -> ForAll (Str_nth_tl m x).
Proof.
induction m.
tauto.
intros x [_ H].
simpl.
apply IHm.
assumption.
Qed.
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 : Type.
Variable f : A -> B.
CoFixpoint map (s:Stream A) : Stream B := Cons (f (hd s)) (map (tl s)).
Lemma Str_nth_tl_map : forall n s, Str_nth_tl n (map s)= map (Str_nth_tl n s).
Proof.
induction n.
reflexivity.
simpl.
intros s.
apply IHn.
Qed.
Lemma Str_nth_map : forall n s, Str_nth n (map s)= f (Str_nth n s).
Proof.
intros n s.
unfold Str_nth.
rewrite Str_nth_tl_map.
reflexivity.
Qed.
Lemma ForAll_map : forall (P:Stream B -> Prop) (S:Stream A), ForAll (fun s => P
(map s)) S <-> ForAll P (map S).
Proof.
intros P S.
split; generalize S; clear S; cofix; intros S; constructor;
destruct H as [H0 H]; firstorder.
Qed.
Lemma Exists_map : forall (P:Stream B -> Prop) (S:Stream A), Exists (fun s => P
(map s)) S -> Exists P (map S).
Proof.
intros P S H.
(induction H;[left|right]); firstorder.
Defined.
End Map.
Section Constant_Stream.
Variable A : Type.
Variable a : A.
CoFixpoint const : Stream A := Cons a const.
End Constant_Stream.
Section Zip.
Variable A B C : Type.
Variable f: A -> B -> C.
CoFixpoint zipWith (a:Stream A) (b:Stream B) : Stream C :=
Cons (f (hd a) (hd b)) (zipWith (tl a) (tl b)).
Lemma Str_nth_tl_zipWith : forall n (a:Stream A) (b:Stream B),
Str_nth_tl n (zipWith a b)= zipWith (Str_nth_tl n a) (Str_nth_tl n b).
Proof.
induction n.
reflexivity.
intros [x xs] [y ys].
unfold Str_nth in *.
simpl in *.
apply IHn.
Qed.
Lemma Str_nth_zipWith : forall n (a:Stream A) (b:Stream B), Str_nth n (zipWith a
b)= f (Str_nth n a) (Str_nth n b).
Proof.
intros.
unfold Str_nth.
rewrite Str_nth_tl_zipWith.
reflexivity.
Qed.
End Zip.
Unset Implicit Arguments.
|
