proposition: Lemma R_pring_as_IR : forall x, RasIR (pring _ x) [=] pring _ x.
proof: Proof. intro x. stepr (nring (R := IR) (nat_of_P x)). stepl (RasIR (nring (R := RReals) (nat_of_P x))). apply R_nring_as_IR. apply R_as_IR_wd. symmetry. apply (pring_convert RReals x). symmetry. apply (pring_convert IR x). Qed.
imports: Coq.QArith.QArith Coq.QArith.Qreals Coq.QArith.QArith_base CoRN.tactics.CornTac Coq.Reals.RIneq Coq.Reals.Rcomplete Coq.Reals.Rlimit Coq.Reals.Rbasic_fun CoRN.coq_reals.Rreals CoRN.reals.iso_CReals CoRN.reals.CauchySeq Coq.Reals.Rtrigo_def CoRN.transc.PowerSeries Coq.Logic.ConstructiveEpsilon Coq.Reals.Rlogic Coq.Reals.Reals CoRN.transc.Pi CoRN.transc.MoreArcTan CoRN.logic.PropDecid CoRN.transc.Exponential Coq(Lia Lra)