CoRN.model.ordfields.Qordfield

Require Export Qfield.
Require Import COrdFields.
Require Import stdlib_omissions.Q.

Definition Qlt_is_strict_order := Build_strictorder
 Qlt_trans Qlt_is_antisymmetric_unfolded.

Definition Q_is_COrdField := Build_is_COrdField Q_as_CField
 Qlt_is_CSetoid_relation Qle (default_greater Q_as_CField Qlt_is_CSetoid_relation)
 (default_grEq Q_as_CField Qle) Qlt_is_strict_order (fun x y E z ⇒ proj2 (Qplus_lt_l x y z) E)
 Qmult_lt_0_compat Qlt_gives_apartness Qle_is_not_lt Qgt_is_lt Qge_is_not_gt.

Definition Q_as_COrdField := Build_COrdField _ _ _ _ _ Q_is_COrdField.

Canonical Structure Q_as_COrdField.

Theorem Q_is_archemaedian : ∀ x : Q_as_COrdField, {n : nat | x [<] nring n}.
Proof.
 intros x. destruct (Q_is_archemaedian0 x) as [n Pn].
 ∃ (nat_of_P n). simpl in ×.
 rewrite nring_Q. rewrite <-Zpos_eq_Z_of_nat_o_nat_of_P. assumption.
Qed.