Library Coq.Vectors.Fin

Require Arith_base.

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.

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.

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.

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.

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.

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).

Definition L_R {m} n (p : t m) : t (n + m).
Defined.

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)).