Library Coq.Vectors.Fin

Require Arith_base.

fin n is a convenient way to represent \1 .. n\

fin n can be seen as a n-uplet of unit. F1 is the first element of the n-uplet. If f is the k-th element of the (n-1)-uplet, FS f is the (k+1)-th element of the n-uplet.

Author: Pierre Boutillier Institution: PPS, INRIA 12/2010-01/2012-07/2012

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 with | F1 | FS _ => fun devil => False_rect (@IDProp) devil end.

Definition caseS' {n : nat} (p : t (S n)) : forall (P : t (S n) -> Type)
  (P1 : P F1) (PS : forall (p : t n), P (FS p)), P p :=
  match p with
  | @F1 k => fun P P1 PS => P1
  | FS pp => fun P P1 PS => PS pp
  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 := caseS' p P (P1 n) PS.

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) {struct a} : forall (b : t n), P a b :=
    match a with
    | @F1 m => fun (b : t (S m)) => caseS' b (P F1) (H0 _) H1
    | @FS m a' => fun (b : t (S m)) =>
      caseS' b (fun b => P (@FS m a') b) (H2 a') (fun b' => HS _ _ (rect2_fix a' b'))
    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 with
    |@F1 j => exist _ 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 _ p eq_refl)
   |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_ext {p}{n} (h h' : p < n) : of_nat_lt h = of_nat_lt h'.

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

Lemma to_nat_of_nat {p}{n} (h : p < n) : to_nat (of_nat_lt h) = exist _ p h.

Lemma to_nat_inj {n} (p q : t n) :
proj1_sig (to_nat p) = proj1_sig (to_nat q) -> p = q.

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).

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 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)).

Fixpoint eqb {m n} (p : t m) (q : t n) :=
match p, q with
| @F1 m', @F1 n' => EqNat.beq_nat m' n'
| FS _, F1 => false
| F1, FS _ => false
| FS p', FS q' => eqb p' q'
end.

Lemma eqb_nat_eq : forall m n (p : t m) (q : t n), eqb p q = true -> m = n.

Lemma eqb_eq : forall n (p q : t n), eqb p q = true <-> p = q.

Lemma eq_dec {n} (x y : t n): {x = y } + {x <> y }.

Definition cast: forall {m} (v: t m) {n}, m = n -> t n.

Library Coq.Vectors.Fin

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