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(* 02110-1301 USA                                                     *)
(** %\subsection*{ support :  finite.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Arith.
Require Export equal_syntax.
(** - The setoid of natural numbers - not really necessary but it's nice to have a uniform approach *)
Definition Nat : Setoid.
apply (Build_Setoid (Equal:=fun i j : nat => i = j)).
split; try split; red in |- *; unfold app_rel in |- *; simpl in |- *; auto.
intros; transitivity y; auto.
Defined.
(** - In this formalisation, we often need the sets $\{0...n-1\}$ We use these for finite sequences: $(a_1...a_n)$ is represented as a setoid function $a:\{0...n-1\}\to A$; also these elements are used as indices into these sequences. Since we use a function type to represent a sequence, we can just type [(a i)] for $a_i$. - finiteT will serve as the Type on which we base our finite setoids [(fin N)] = $\{0...N-1\} = \{ n \in N | n<N \}$. *)
Record finiteT (N : Nat) : Type :=  {index : nat; in_range_prf : index < N}.
(** - The setoid itself is called [(fin N)]. [n:(finiteT N)] is really a pair <n,H>; <n,H> and <n',H'> will be deemed equal if n and n' are; the proofs H of n<N and H' of n'(=n)<N may differ.  - %\bf% Useful knowledge: if [H:(lt i n)] then [(Build_finiteT H):(fin n)]  %\label{fin}% *)
Definition fin : Nat -> Setoid.
intro N.
apply
(Build_Setoid (Carrier:=finiteT N)
(Equal:=fun n m : finiteT N => index n = index m)).
(* Now, we prove that our relation is an equivalence *)
red in |- *.
split.
{ (* Reflexivity *)
red in |- *.
intro.
red in |- *.
reflexivity.
}
{ (* Partial equivalence *)
red in |- *.
split.
{ (* Transitivity *)
red in |- *.
intros.
red in H.
red in H0.
red in |- *.
transitivity (index y); assumption.
}
{ (* Symmetry *)
red in |- *.
intros.
red in H.
red in |- *.
symmetry  in |- *.
assumption.
}
}
Defined.
intros.
simpl in |- *.
auto.
Qed.
(* This formalisation is heavily dependent on Loic Pottier's algebra contribution *)
(* and therefore relies heavily on the algebra Hints database as well. *)
(* We feel free to extend it: *)
Hint Resolve fin_equation: algebra.
intros.
simpl in |- *.
cut (forall k l : nat, k = l \/ k <> l).
intro.
auto.
clear i' i n.
double induction k l.
left; auto.
intros; right.
unfold not in |- *.
intro.
inversion H0.
intros; right.
unfold not in |- *.
intro.
inversion H0.
intros.
elim (H0 n).
auto.
auto.
Qed.
destruct 1.
inversion_clear in_range_prf0.
Qed.
Hint Resolve fin_O_nonexistent: algebra.

Lemma fin_S_O_unique : forall i j : fin 1, i =' j in _.