CoRN.model.setoids.Zfinsetoid

Require Export ZArith.
Require Import CSetoids.
Set Automatic Introduction.

Record ZF (n:Z):Set:=
{ZF_crr:> Z ;
ZF_prf0: (Zlt ZF_crr n);
ZF_prf1: (Zle 0 ZF_crr)
}.

Definition ZFeq (n : Z) : ZF n → ZF n → Prop.
Proof.
 intros a b.
 case a.
 case b.
 intros x H H' x0 H0 H0'.
 exact (x = x0).
Defined.

Definition ZFap (n : Z) : ZF n → ZF n → CProp.
Proof.
 intros a b.
 case a.
 case b.
 intros x H H' x0 H0 H0'.
 exact (x ≠ x0).
Defined.

Lemma ZFap_irreflexive : ∀ n : Z, irreflexive (ZFap n).
Proof.
 unfold irreflexive in |- ×.
 unfold ZFap in |- ×.
 intros n x.
 case x.
 intuition.
 red in |- ×.
 intuition.
Qed.

Lemma ZFap_symmetric : ∀ n : Z, Csymmetric (ZFap n).
Proof.
 intro n.
 unfold Csymmetric in |- ×.
 unfold ZFap in |- ×.
 intros x y.
 case x.
 case y.
 intuition.
Qed.

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

Lemma ZFap_tight : ∀ n : Z, tight_apart (ZFeq n) (ZFap n).
Proof.
 unfold tight_apart in |- ×.
 unfold ZFap in |- ×.
 unfold ZFeq in |- ×.
 intros n x y.
 case x.
 case y.
 intros x0 H0 H0'x1 H1 H1'.
 red in |- ×.
 unfold not in |- ×.
 unfold Not in |- ×.
 intuition.
Qed.

Definition Zless (n : Z) :=
  Build_is_CSetoid (ZF n) (ZFeq n) (ZFap n) (ZFap_irreflexive n)
    (ZFap_symmetric n) (ZFap_cotransitive n) (ZFap_tight n).

Definition ZCSetoid_of_less (n : Z) : CSetoid :=
  Build_CSetoid (ZF n) (ZFeq n) (ZFap n) (Zless n).