CoRN.model.setoids.Nfinsetoid

Require Import CSetoids.
Set Automatic Introduction.

Record F (n:nat):Set:=
{F_crr:> nat ;
F_prf: F_crr<n
}.

Definition Feq (n : nat) : F n → F n → Prop.
Proof.
 intros a b.
 case a.
 case b.
 intros x H x0 H0.
 exact (x = x0).
Defined.

Definition Fap (n : nat) : F n → F n → CProp.
Proof.
 intros a b.
 case a.
 case b.
 intros x H x0 H0.
 exact (x ≠ x0).
Defined.

Lemma Fap_irreflexive : ∀ n : nat, irreflexive (Fap n).
Proof.
 unfold irreflexive in |- ×.
 unfold Fap in |- ×.
 intros n x.
 case x.
 intuition.
 red in |- ×.
 intuition.
Qed.

Lemma Fap_symmetric : ∀ n : nat, Csymmetric (Fap n).
Proof.
 intro n.
 unfold Csymmetric in |- ×.
 unfold Fap in |- ×.
 intros x y.
 case x.
 case y.
 intuition.
Qed.

Lemma Fap_cotransitive : ∀ n : nat, cotransitive (Fap n).
Proof.
 intro n.
 unfold cotransitive in |- ×.
 unfold Fap in |- ×.
 intros x y.
 case x.
 case y.
 intros x0 H0 x1 H1 H2 z.
 case z.
 intros x2 H.
 set (H5 := eq_nat_dec x2 x1) in ×.
 elim H5.
  clear H5.
  intro H5.
  right.
  rewrite H5.
  exact H2.
 clear H5.
 intro H5.
 left.
 exact H5.
Qed.

Lemma Fap_tight : ∀ n : nat, tight_apart (Feq n) (Fap n).
Proof.
 unfold tight_apart in |- ×.
 unfold Fap in |- ×.
 unfold Feq in |- ×.
 intros n x y.
 case x.
 case y.
 intros x0 H0 x1 H1.
 red in |- ×.
 unfold not in |- ×.
 unfold Not in |- ×.
 intuition.
Qed.

Definition less (n : nat) :=
  Build_is_CSetoid (F n) (Feq n) (Fap n) (Fap_irreflexive n)
    (Fap_symmetric n) (Fap_cotransitive n) (Fap_tight n).

Definition CSetoid_of_less (n : nat) : CSetoid :=
  Build_CSetoid (F n) (Feq n) (Fap n) (less n).