Section SCHEMES.
Definition case0 P (p: t 0): P p :=
match p with | F1 | FS  _ => fun devil => False_rect (@IDProp) devil (* subterm !!! *) 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.
(** [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.
(** The p{^ th} element of [fin m] viewed as the p{^ th} element of [fin (n + m)] Really really inefficient !!! *)
Definition L_R {m} n (p : t m) : t (n + m).
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.
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.