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
|
Require Import ExtLib.Data.Fin.
Set Implicit Arguments.
Set Strict Implicit.
Set Asymmetric Patterns.
Inductive vector T : nat -> Type :=
| Vnil : vector T 0
| Vcons : forall {n}, T -> vector T n -> vector T (S n).
Section parametric.
Variable T : Type.
Definition vector_hd n (v : vector T (S n)) : T :=
match v in vector _ n' return match n' with
| 0 => unit
| S _ => T
end with
| Vnil => tt
| Vcons _ x _ => x
end.
Definition vector_tl n (v : vector T (S n)) : vector T n :=
match v in vector _ n' return match n' with
| 0 => unit
| S n => vector T n
end with
| Vnil => tt
| Vcons _ _ x => x
end.
Theorem vector_eta : forall n (v : vector T n),
v = match n as n return vector T n -> vector T n with
| 0 => fun _ => Vnil _
| S n => fun v => Vcons (vector_hd v) (vector_tl v)
end v.
Proof.
destruct v; auto.
Qed.
Fixpoint get {n : nat} (f : fin n) : vector T n -> T :=
match f in fin n return vector T n -> T with
| F0 n => @vector_hd _
| FS n f => fun v => get f (vector_tl v)
end.
Fixpoint put {n : nat} (f : fin n) (t : T) : vector T n -> vector T n :=
match f in fin n return vector T n -> vector T n with
| F0 _ => fun v => Vcons t (vector_tl v)
| FS _ f => fun v => Vcons (vector_hd v) (put f t (vector_tl v))
end.
Theorem get_put_eq : forall {n} (v : vector T n) (f : fin n) val,
get f (put f val v) = val.
Proof.
induction n.
{ inversion f. }
{ remember (S n). destruct f.
inversion Heqn0; subst; intros; reflexivity.
inversion Heqn0; subst; simpl; auto. }
Qed.
Theorem get_put_neq : forall {n} (v : vector T n) (f f' : fin n) val,
f <> f' ->
get f (put f' val v) = get f v.
Proof.
induction n.
{ inversion f. }
{ remember (S n); destruct f.
{ inversion Heqn0; clear Heqn0; subst; intros.
destruct (fin_case f'); try congruence.
destruct H0; subst. auto. }
{ inversion Heqn0; clear Heqn0; subst; intros.
destruct (fin_case f').
subst; auto.
destruct H0; subst. simpl.
eapply IHn. congruence. } }
Qed.
Section ForallV.
Variable P : T -> Prop.
Inductive ForallV : forall n, vector T n -> Prop :=
| ForallV_nil : ForallV (Vnil _)
| ForallV_cons : forall n e es, P e -> @ForallV n es -> ForallV (Vcons e es).
Definition ForallV_vector_hd n (v : vector T (S n)) (f : ForallV v) : P (vector_hd v) :=
match f in @ForallV n v return match n as n return vector T n -> Prop with
| 0 => fun _ => True
| S _ => fun v => P (vector_hd v)
end v
with
| ForallV_nil => I
| ForallV_cons _ _ _ pf _ => pf
end.
Definition ForallV_vector_tl n (v : vector T (S n)) (f : ForallV v) : ForallV (vector_tl v) :=
match f in @ForallV n v return match n as n return vector T n -> Prop with
| 0 => fun _ => True
| S _ => fun v => ForallV (vector_tl v)
end v
with
| ForallV_nil => I
| ForallV_cons _ _ _ _ pf => pf
end.
End ForallV.
Section vector_dec.
Variable Tdec : forall a b : T, {a = b} + {a <> b}.
Fixpoint vector_dec {n} (a : vector T n)
: forall b : vector T n, {a = b} + {a <> b} :=
match a in vector _ n
return forall b : vector T n, {a = b} + {a <> b}
with
| Vnil => fun b => left match b in vector _ 0 with
| Vnil => eq_refl
end
| Vcons _ a a' => fun b =>
match b as b in vector _ (S n)
return forall a',
(forall a : vector T n, {a' = a} + {a' <> a}) ->
{Vcons a a' = b} + {Vcons a a' <> b}
with
| Vcons _ b b' => fun a' rec =>
match Tdec a b , rec b' with
| left pf , left pf' =>
left match pf , pf' with
| eq_refl , eq_refl => eq_refl
end
| right pf , _ =>
right (fun x : Vcons a a' = Vcons b b' =>
pf match x in _ = z
return a = vector_hd z
with
| eq_refl => eq_refl
end)
| left _ , right pf =>
right (fun x : Vcons a a' = Vcons b b' =>
pf match x in _ = z
return a' = vector_tl z
with
| eq_refl => eq_refl
end)
end
end a' (@vector_dec _ a')
end.
End vector_dec.
Section vector_in.
Variable a : T.
Inductive vector_In : forall {n}, vector T n -> Prop :=
| vHere : forall n rst, @vector_In (S n) (Vcons a rst)
| vNext : forall n rst b, @vector_In n rst ->
@vector_In (S n) (Vcons b rst).
End vector_in.
Lemma ForallV_vector_In : forall {n} t (vs : vector T n) P,
ForallV P vs ->
vector_In t vs -> P t.
Proof.
induction 2.
- eapply (ForallV_vector_hd H).
- eapply IHvector_In. eapply (ForallV_vector_tl H).
Qed.
End parametric.
Section vector_map.
Context {T U : Type}.
Variable f : T -> U.
Fixpoint vector_map {n} (v : vector T n) : vector U n :=
match v with
| Vnil => Vnil _
| Vcons _ v vs => Vcons (f v) (vector_map vs)
end.
End vector_map.
Arguments vector T n.
Arguments vector_hd {T n} _.
Arguments vector_tl {T n} _.
|
