Variable n : nat.
Inductive ordinalGI : predArgType := OrdinalGI x of ('N x < n)%N.
Coercion GI_of_ord i := let: OrdinalGI m _ := i in m.
Canonical ordinalGI_subType := [subType for GI_of_ord].
Definition ordinalGI_eqMixin := Eval hnf in [eqMixin of ordinalGI by <:].
Canonical ordinalGI_eqType := Eval hnf in EqType ordinalGI ordinalGI_eqMixin.
Definition ordinalGI_choiceMixin := [choiceMixin of ordinalGI by <:].
Canonical ordinalGI_choiceType :=
Eval hnf in ChoiceType ordinalGI ordinalGI_choiceMixin.
Definition ordinalGI_countMixin := [countMixin of ordinalGI by <:].
Canonical ordinalGI_countType :=
Eval hnf in CountType ordinalGI ordinalGI_countMixin.
Canonical ordinalGI_subCountType := [subCountType of ordinalGI].

Lemma ltn_ordGI (i : ordinalGI) : ('N i < n)%N.
Fact int_norm_nat x y m :
x \in Cint ->  y \in Cint ->
`|x| ^+ 2 + `|y| ^+ 2 < m%:R -> (x + m%:R \in Cnat) && (x + m%:R < m.*2%:R).