# Library Coq.Vectors.VectorDef

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.
Open Local Scope nat_scope.

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
|nil => fun devil => False_rect (@ID) devil
|cons a 0 v =>
match v as vnn in t _ nn
return
match nn,vnn with
|0,vm => P (a :: vm)
|S _,_ => _
end
with
|nil => bas a
|_ :: _ => fun devil => False_rect (@ID) devil
end
|cons a (S nn') v => rect a v (rectS_fix v)
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 as v1' in t _ n1
return forall v2 : t B n1, P v1' v2 with
|[] => fun v2 =>
match v2 with
|[] => bas
|_ :: _ => fun devil => False_rect (@ID) devil
end
|h1 :: t1 => fun v2 =>
match v2 with
|[] => fun devil => False_rect (@ID) devil
|h2 :: t2 => fun t1' =>
rect (rect2_fix t1' t2) h1 h2
end t1
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
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 as v' in t _ m return match m, v' with |0, _ => False -> True |S _, v0 => P v' end with
|[] => fun devil => False_rect _ devil
|h :: t => H h t
end.
End SCHEMES.

Section BASES.
The first element of a non empty vector
Definition hd {A} {n} (v:t A (S n)) := Eval cbv delta beta in
(caseS (fun n v => A) (fun h n t => h) v).

The last element of an non empty vector
Definition last {A} {n} (v : t A (S n)) := Eval cbv delta in
(rectS (fun _ _ => A) (fun a => a) (fun _ _ _ H => H) v).

Build a vector of n{^ th} a
Fixpoint const {A} (a:A) (n:nat) :=
match n return t A n with
| O => nil A
| S n => a :: (const a n)
end.

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 q => fun v => caseS (fun n v' => A) (fun h n t => h) v
|Fin.FS q 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 (fun n _ => t A (S n)) (fun h _ t => a :: t) v'
|Fin.FS k p' => fun v' =>
(caseS (fun n _ => Fin.t n -> t A (S n)) (fun h _ t p2 => h :: (replace t p2 a)) v') p'
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} {n} (v:t A (S n)) := Eval cbv delta beta in
(caseS (fun n v => t A n) (fun h n t => t) v).

Remove last element of a non-empty vector
Definition shiftout {A} {n:nat} (v:t A (S n)) : t A n :=
Eval cbv delta beta in (rectS (fun n _ => t A n) (fun a => [])
(fun h _ _ H => h :: H) 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} {n} (v:t A (S n)) : t A (S (S n)) :=
Eval cbv delta beta in (rectS (fun n _ => t A (S (S n)))
(fun h => h :: h :: []) (fun h _ _ H => h :: H) v).

Remove p last elements of a vector
Lemma trunc : forall {A} {n} (p:nat), n > p -> t A n
-> t A (n - p).

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) {n} (v1:t A n) (v2:t B n)
: t C n :=
Eval cbv delta beta in rect2 (fun n _ _ => t C n) (nil C)
(fun _ _ _ H a b => (g a b) :: H) 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 x1 .. xn y1 .. yn c = g x1 y1 (g x2 y2 .. (g xn yn c) .. )
Definition fold_right2 {A B C} (g:A -> B -> C -> C) {n} (v:t A n)
(w : t B n) (c:C) : C :=
Eval cbv delta beta in rect2 (fun _ _ _ => C) c
(fun _ _ _ H a b => g a b H) v w.

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 => match w in t _ n1
return match n1 with |0 => A |S _ => @ID end with
|[] => a
|_ :: _ => @id end
|cons vh vn vt => fun w => match w in t _ n1
return match n1 with |0 => @ID |S n => t B n -> A end with
|[] => @id
|wh :: wt => fun vt' => fold_left2_fix (f a vh wh) vt' wt end vt
end.

End ITERATORS.

Section SCANNING.
Inductive Forall {A} (P: A -> Prop): forall {n} (v: t A n), Prop :=
|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.