# Library Coq.Vectors.Fin

Require Arith_base.

fin n is a convinient way to represent \1 .. n\
fin n can be seen as a n-uplet of unit where F1 is the first element of the n-uplet and FS set (n-1)-uplet of all the element but the first.
Author: Pierre Boutillier Institution: PPS, INRIA 12/2010-01/2012

Inductive t : nat -> Set :=
|F1 : forall {n}, t (S n)
|FS : forall {n}, t n -> t (S n).

Section SCHEMES.
Definition case0 P (p: t 0): P p :=
match p as p' in t n return
match n as n' return t n' -> Type
with |0 => fun f0 => P f0 |S _ => fun _ => @ID end p'
with |F1 _ => @id |FS _ _ => @id end.

Definition caseS (P: forall {n}, t (S n) -> Type)
(P1: forall n, @P n F1) (PS : forall {n} (p: t n), P (FS p))
{n} (p: t (S n)): P p :=
match p with
|F1 k => P1 k
|FS k pp => PS pp
end.

Definition rectS (P: forall {n}, t (S n) -> Type)
(P1: forall n, @P n F1) (PS : forall {n} (p: t (S n)), P p -> P (FS p)):
forall {n} (p: t (S n)), P p :=
fix rectS_fix {n} (p: t (S n)): P p:=
match p with
|F1 k => P1 k
|FS 0 pp => case0 (fun f => P (FS f)) pp
|FS (S k) pp => PS pp (rectS_fix pp)
end.

Definition rect2 (P: forall {n} (a b: t n), Type)
(H0: forall n, @P (S n) F1 F1)
(H1: forall {n} (f: t n), P F1 (FS f))
(H2: forall {n} (f: t n), P (FS f) F1)
(HS: forall {n} (f g : t n), P f g -> P (FS f) (FS g)):
forall {n} (a b: t n), P a b :=
fix rect2_fix {n} (a: t n): forall (b: t n), P a b :=
match a with
|F1 m => fun (b: t (S m)) => match b as b' in t n'
return match n',b' with
|0,_ => @ID
|S n0,b0 => P F1 b0
end with
|F1 m' => H0 m'
|FS m' b' => H1 b'
end
|FS m a' => fun (b: t (S m)) => match b with
|F1 m' => fun aa: t m' => H2 aa
|FS m' b' => fun aa: t m' => HS aa b' (rect2_fix aa b')
end a'
end.
End SCHEMES.

Definition FS_inj {n} (x y: t n) (eq: FS x = FS y): x = y :=
match eq in _ = a return
match a as a' in t m return match m with |0 => Prop |S n' => t n' -> Prop end
with @F1 _ => fun _ => True |@FS _ y => fun x' => x' = y end x with
eq_refl => eq_refl
end.

to_nat f = p iff f is the p{^ th} element of fin m.
Fixpoint to_nat {m} (n : t m) : {i | i < m} :=
match n in t k return {i | i< k} with
|F1 j => exist (fun i => i< S j) 0 (Lt.lt_0_Sn j)
|FS _ p => match to_nat p with |exist i P => exist _ (S i) (Lt.lt_n_S _ _ P) end
end.

of_nat p n answers the p{^ th} element of fin n if p < n or a proof of p >= n else
Fixpoint of_nat (p n : nat) : (t n) + { exists m, p = n + m } :=
match n with
|0 => inright _ (ex_intro (fun x => p = 0 + x) p (@eq_refl _ p))
|S n' => match p with
|0 => inleft _ (F1)
|S p' => match of_nat p' n' with
|inleft f => inleft _ (FS f)
|inright arg => inright _ (match arg with |ex_intro m e =>
ex_intro (fun x => S p' = S n' + x) m (f_equal S e) end)
end
end
end.

of_nat_lt p n H answers the p{^ th} element of fin n it behaves much better than of_nat p n on open term
Fixpoint of_nat_lt {p n : nat} : p < n -> t n :=
match n with
|0 => fun H : p < 0 => False_rect _ (Lt.lt_n_O p H)
|S n' => match p with
|0 => fun _ => @F1 n'
|S p' => fun H => FS (of_nat_lt (Lt.lt_S_n _ _ H))
end
end.

Lemma of_nat_to_nat_inv {m} (p : t m) : of_nat_lt (proj2_sig (to_nat p)) = p.

weak p f answers a function witch is the identity for the p{^ th} first element of fin (p + m) and FS (FS .. (FS (f k))) for FS (FS .. (FS k)) with p FSs
Fixpoint weak {m}{n} p (f : t m -> t n) :
t (p + m) -> t (p + n) :=
match p as p' return t (p' + m) -> t (p' + n) with
|0 => f
|S p' => fun x => match x with
|F1 n' => fun eq : n' = p' + m => F1
|FS n' y => fun eq : n' = p' + m => FS (weak p' f (eq_rect _ t y _ eq))
end (eq_refl _)
end.

The p{^ th} element of fin m viewed as the p{^ th} element of fin (m + n)
Fixpoint L {m} n (p : t m) : t (m + n) :=
match p with |F1 _ => F1 |FS _ p' => FS (L n p') end.

Lemma L_sanity {m} n (p : t m) : proj1_sig (to_nat (L n p)) = proj1_sig (to_nat p).

The p{^ th} element of fin m viewed as the p{^ th} element of fin (n + m) Really really ineficient !!!
Definition L_R {m} n (p : t m) : t (n + m).
Defined.

The p{^ th} element of fin m viewed as the (n + p){^ th} element of fin (n + m)
Fixpoint R {m} n (p : t m) : t (n + m) :=
match n with |0 => p |S n' => FS (R n' p) end.

Lemma R_sanity {m} n (p : t m) : proj1_sig (to_nat (R n p)) = n + proj1_sig (to_nat p).

Fixpoint depair {m n} (o : t m) (p : t n) : t (m * n) :=
match o with
|F1 m' => L (m' * n) p
|FS m' o' => R n (depair o' p)
end.

Lemma depair_sanity {m n} (o : t m) (p : t n) :
proj1_sig (to_nat (depair o p)) = n * (proj1_sig (to_nat o)) + (proj1_sig (to_nat p)).