-
Notifications
You must be signed in to change notification settings - Fork 1
/
Vector.v
312 lines (251 loc) · 10.1 KB
/
Vector.v
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
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** Definitions of Vectors and functions to use them
Author: Pierre Boutillier
Institution: PPS, INRIA 12/2010
*)
(**
Names should be "caml name in list.ml" if exists and order of arguments
have to be the same. complain if you see mistakes ... *)
Require Import Arith_base.
Require Vectors.Fin.
Import EqNotations.
Local Open Scope nat_scope.
(* Duplicated to set Universe Polymorphism. *)
Set Universe Polymorphism.
(**
A vector is a list of size n whose elements belong to a set A. *)
Inductive t A : nat -> Type :=
|nil : t A 0
|cons : forall (h:A) (n:nat), t A n -> t A (S n).
Local Notation "[]" := (nil _).
Local Notation "h :: t" := (cons _ h _ t) (at level 60, right associativity).
Section SCHEMES.
(** An induction scheme for non-empty vectors *)
Definition rectS {A} (P:forall {n}, t A (S n) -> Type)
(bas: forall a: A, P (a :: []))
(rect: forall a {n} (v: t A (S n)), P v -> P (a :: v)) :=
fix rectS_fix {n} (v: t A (S n)) : P v :=
match v with
|@cons _ a 0 v =>
match v with
|nil _ => bas a
|_ => fun devil => False_ind (@IDProp) devil (* subterm !!! *)
end
|@cons _ a (S nn') v => rect a v (rectS_fix v)
|_ => fun devil => False_ind (@IDProp) devil (* subterm !!! *)
end.
(** A vector of length [0] is [nil] *)
Definition case0 {A} (P:t A 0 -> Type) (H:P (nil A)) v:P v :=
match v with
|[] => H
|_ => fun devil => False_ind (@IDProp) devil (* subterm !!! *)
end.
(** A vector of length [S _] is [cons] *)
Definition caseS {A} (P : forall {n}, t A (S n) -> Type)
(H : forall h {n} t, @P n (h :: t)) {n} (v: t A (S n)) : P v :=
match v with
|h :: t => H h t
|_ => fun devil => False_ind (@IDProp) devil (* subterm !!! *)
end.
Definition caseS' {A} {n : nat} (v : t A (S n)) : forall (P : t A (S n) -> Type)
(H : forall h t, P (h :: t)), P v :=
match v with
| h :: t => fun P H => H h t
| _ => fun devil => False_rect (@IDProp) devil
end.
(** An induction scheme for 2 vectors of same length *)
Definition rect2 {A B} (P:forall {n}, t A n -> t B n -> Type)
(bas : P [] []) (rect : forall {n v1 v2}, P v1 v2 ->
forall a b, P (a :: v1) (b :: v2)) :=
fix rect2_fix {n} (v1 : t A n) : forall v2 : t B n, P v1 v2 :=
match v1 with
| [] => fun v2 => case0 _ bas v2
| @cons _ h1 n' t1 => fun v2 =>
caseS' v2 (fun v2' => P (h1::t1) v2') (fun h2 t2 => rect (rect2_fix t1 t2) h1 h2)
end.
End SCHEMES.
Section BASES.
(** The first element of a non empty vector *)
Definition hd {A} := @caseS _ (fun n v => A) (fun h n t => h).
Global Arguments hd {A} {n} v.
(** The last element of an non empty vector *)
Definition last {A} := @rectS _ (fun _ _ => A) (fun a => a) (fun _ _ _ H => H).
Global Arguments last {A} {n} v.
(** Build a vector of n{^ th} [a] *)
Definition const {A} (a:A) := nat_rect _ [] (fun n x => cons _ a n x).
(** The [p]{^ th} element of a vector of length [m].
Computational behavior of this function should be the same as
ocaml function. *)
Definition nth {A} :=
fix nth_fix {m} (v' : t A m) (p : Fin.t m) {struct v'} : A :=
match p in Fin.t m' return t A m' -> A with
|Fin.F1 => caseS (fun n v' => A) (fun h n t => h)
|Fin.FS p' => fun v => (caseS (fun n v' => Fin.t n -> A)
(fun h n t p0 => nth_fix t p0) v) p'
end v'.
(** An equivalent definition of [nth]. *)
Definition nth_order {A} {n} (v: t A n) {p} (H: p < n) :=
(nth v (Fin.of_nat_lt H)).
(** Put [a] at the p{^ th} place of [v] *)
Fixpoint replace {A n} (v : t A n) (p: Fin.t n) (a : A) {struct p}: t A n :=
match p with
| @Fin.F1 k => fun v': t A (S k) => caseS' v' _ (fun h t => a :: t)
| @Fin.FS k p' => fun v' : t A (S k) =>
(caseS' v' (fun _ => t A (S k)) (fun h t => h :: (replace t p' a)))
end v.
(** Version of replace with [lt] *)
Definition replace_order {A n} (v: t A n) {p} (H: p < n) :=
replace v (Fin.of_nat_lt H).
(** Remove the first element of a non empty vector *)
Definition tl {A} := @caseS _ (fun n v => t A n) (fun h n t => t).
Global Arguments tl {A} {n} v.
(** Remove last element of a non-empty vector *)
Definition shiftout {A} := @rectS _ (fun n _ => t A n) (fun a => [])
(fun h _ _ H => h :: H).
Global Arguments shiftout {A} {n} v.
(** Add an element at the end of a vector *)
Fixpoint shiftin {A} {n:nat} (a : A) (v:t A n) : t A (S n) :=
match v with
|[] => a :: []
|h :: t => h :: (shiftin a t)
end.
(** Copy last element of a vector *)
Definition shiftrepeat {A} := @rectS _ (fun n _ => t A (S (S n)))
(fun h => h :: h :: []) (fun h _ _ H => h :: H).
Global Arguments shiftrepeat {A} {n} v.
(** Remove [p] last elements of a vector *)
Lemma trunc : forall {A} {n} (p:nat), n > p -> t A n
-> t A (n - p).
Proof.
induction p as [| p f]; intros H v.
rewrite <- minus_n_O.
exact v.
apply shiftout.
rewrite minus_Sn_m.
apply f.
auto with *.
exact v.
auto with *.
Defined.
(** Concatenation of two vectors *)
Fixpoint append {A}{n}{p} (v:t A n) (w:t A p):t A (n+p) :=
match v with
| [] => w
| a :: v' => a :: (append v' w)
end.
Infix "++" := append.
(** Two definitions of the tail recursive function that appends two lists but
reverses the first one *)
(** This one has the exact expected computational behavior *)
Fixpoint rev_append_tail {A n p} (v : t A n) (w: t A p)
: t A (tail_plus n p) :=
match v with
| [] => w
| a :: v' => rev_append_tail v' (a :: w)
end.
Import EqdepFacts.
(** This one has a better type *)
Definition rev_append {A n p} (v: t A n) (w: t A p)
:t A (n + p) :=
rew <- (plus_tail_plus n p) in (rev_append_tail v w).
(** rev [a₁ ; a₂ ; .. ; an] is [an ; a{n-1} ; .. ; a₁]
Caution : There is a lot of rewrite garbage in this definition *)
Definition rev {A n} (v : t A n) : t A n :=
rew <- (plus_n_O _) in (rev_append v []).
End BASES.
Local Notation "v [@ p ]" := (nth v p) (at level 1).
Section ITERATORS.
(** * Here are special non dependent useful instantiation of induction
schemes *)
(** Uniform application on the arguments of the vector *)
Definition map {A} {B} (f : A -> B) : forall {n} (v:t A n), t B n :=
fix map_fix {n} (v : t A n) : t B n := match v with
| [] => []
| a :: v' => (f a) :: (map_fix v')
end.
(** map2 g [x1 .. xn] [y1 .. yn] = [(g x1 y1) .. (g xn yn)] *)
Definition map2 {A B C} (g:A -> B -> C) :
forall (n : nat), t A n -> t B n -> t C n :=
@rect2 _ _ (fun n _ _ => t C n) (nil C) (fun _ _ _ H a b => (g a b) :: H).
Global Arguments map2 {A B C} g {n} v1 v2.
(** fold_left f b [x1 .. xn] = f .. (f (f b x1) x2) .. xn *)
Definition fold_left {A B:Type} (f:B->A->B): forall (b:B) {n} (v:t A n), B :=
fix fold_left_fix (b:B) {n} (v : t A n) : B := match v with
| [] => b
| a :: w => (fold_left_fix (f b a) w)
end.
(** fold_right f [x1 .. xn] b = f x1 (f x2 .. (f xn b) .. ) *)
Definition fold_right {A B : Type} (f : A->B->B) :=
fix fold_right_fix {n} (v : t A n) (b:B)
{struct v} : B :=
match v with
| [] => b
| a :: w => f a (fold_right_fix w b)
end.
(** fold_right2 g c [x1 .. xn] [y1 .. yn] = g x1 y1 (g x2 y2 .. (g xn yn c) .. )
c is before the vectors to be compliant with "refolding". *)
Definition fold_right2 {A B C} (g:A -> B -> C -> C) (c: C) :=
@rect2 _ _ (fun _ _ _ => C) c (fun _ _ _ H a b => g a b H).
(** fold_left2 f b [x1 .. xn] [y1 .. yn] = g .. (g (g a x1 y1) x2 y2) .. xn yn *)
Definition fold_left2 {A B C: Type} (f : A -> B -> C -> A) :=
fix fold_left2_fix (a : A) {n} (v : t B n) : t C n -> A :=
match v in t _ n0 return t C n0 -> A with
|[] => fun w => case0 (fun _ => A) a w
|@cons _ vh vn vt => fun w =>
caseS' w (fun _ => A) (fun wh wt => fold_left2_fix (f a vh wh) vt wt)
end.
End ITERATORS.
Section SCANNING.
Inductive Forall {A} (P: A -> Type): forall {n} (v: t A n), Type :=
|Forall_nil: Forall P []
|Forall_cons {n} x (v: t A n): P x -> Forall P v -> Forall P (x::v).
Hint Constructors Forall.
Inductive Exists {A} (P:A->Prop): forall {n}, t A n -> Prop :=
|Exists_cons_hd {m} x (v: t A m): P x -> Exists P (x::v)
|Exists_cons_tl {m} x (v: t A m): Exists P v -> Exists P (x::v).
Hint Constructors Exists.
Inductive In {A} (a:A): forall {n}, t A n -> Prop :=
|In_cons_hd {m} (v: t A m): In a (a::v)
|In_cons_tl {m} x (v: t A m): In a v -> In a (x::v).
Hint Constructors In.
Inductive Forall2 {A B} (P:A->B->Prop): forall {n}, t A n -> t B n -> Prop :=
|Forall2_nil: Forall2 P [] []
|Forall2_cons {m} x1 x2 (v1:t A m) v2: P x1 x2 -> Forall2 P v1 v2 ->
Forall2 P (x1::v1) (x2::v2).
Hint Constructors Forall2.
Inductive Exists2 {A B} (P:A->B->Prop): forall {n}, t A n -> t B n -> Prop :=
|Exists2_cons_hd {m} x1 x2 (v1: t A m) (v2: t B m): P x1 x2 -> Exists2 P (x1::v1) (x2::v2)
|Exists2_cons_tl {m} x1 x2 (v1:t A m) v2: Exists2 P v1 v2 -> Exists2 P (x1::v1) (x2::v2).
Hint Constructors Exists2.
End SCANNING.
Section VECTORLIST.
(** * vector <=> list functions *)
Fixpoint of_list {A} (l : list A) : t A (length l) :=
match l as l' return t A (length l') with
|Datatypes.nil => []
|(h :: tail)%list => (h :: (of_list tail))
end.
Definition to_list {A}{n} (v : t A n) : list A :=
Eval cbv delta beta in fold_right (fun h H => Datatypes.cons h H) v Datatypes.nil.
End VECTORLIST.
Module VectorNotations.
Notation "[]" := [] : vector_scope.
Notation "h :: t" := (h :: t) (at level 60, right associativity)
: vector_scope.
Notation " [ x ] " := (x :: []) : vector_scope.
Notation " [ x ; .. ; y ] " := (cons _ x _ .. (cons _ y _ (nil _)) ..) : vector_scope
.
Notation "v [@ p ]" := (nth v p) (at level 1, format "v [@ p ]") : vector_scope.
Open Scope vector_scope.
End VectorNotations.
Definition cons_inj {A} {a1 a2} {n} {v1 v2 : t A n}
(eq : a1 :: v1 = a2 :: v2) : a1 = a2 /\ v1 = v2 :=
match eq in _ = x return caseS _ (fun a2' _ v2' => fun v1' => a1 = a2' /\ v1' = v2') x v1
with | eq_refl => conj eq_refl eq_refl
end.