# Library Coq.setoid_ring.RealField

Require Import Nnat.
Require Import ArithRing.
Require Export Ring Field.
Require Import Rdefinitions.
Require Import Rpow_def.
Require Import Raxioms.
Open Local Scope R_scope.
Lemma RTheory : ring_theory 0 1 Rplus Rmult Rminus Ropp (eq (A:=R)).
Lemma Rfield : field_theory 0 1 Rplus Rmult Rminus Ropp Rdiv Rinv (eq(A:=R)).
Lemma Rlt_n_Sn : forall x, x < x + 1.
Notation Rset := (Eqsth R).
Notation Rext := (Eq_ext Rplus Rmult Ropp).
Lemma Rlt_0_2 : 0 < 2.
Lemma Rgen_phiPOS : forall x, InitialRing.gen_phiPOS1 1 Rplus Rmult x > 0.
Lemma Rgen_phiPOS_not_0 :
forall x, InitialRing.gen_phiPOS1 1 Rplus Rmult x <> 0.
Lemma Zeq_bool_complete : forall x y,
InitialRing.gen_phiZ 0%R 1%R Rplus Rmult Ropp x =
InitialRing.gen_phiZ 0%R 1%R Rplus Rmult Ropp y ->
Zeq_bool x y = true.
Lemma Rdef_pow_add : forall (x:R) (n m:nat), pow x (n + m) = pow x n * pow x m.
Lemma R_power_theory : power_theory 1%R Rmult (eq (A:=R)) nat_of_N pow.
Ltac Rpow_tac t :=
match isnatcst t with
| false => constr:(InitialRing.NotConstant)
| _ => constr:(N_of_nat t)
end.
Add Field RField : Rfield
(completeness Zeq_bool_complete, power_tac R_power_theory [Rpow_tac]).