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.

Local Open 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 R) N.to_nat pow.

Ltac Rpow_tac t :=
  match isnatcst t with
  | false => constr:(InitialRing.NotConstant)
  | _ => constr:(N.of_nat t)
  end.

Ltac IZR_tac t :=
  match t with
  | R0 => constr:(0%Z)
  | R1 => constr:(1%Z)
  | IZR (Z.pow_pos 10 ?p) =>
    match isPcst p with
    | true => constr:(Z.pow_pos 10 p)
    | _ => constr:(InitialRing.NotConstant)
    end
  | IZR ?u =>
    match isZcst u with
    | true => u
    | _ => constr:(InitialRing.NotConstant)
    end
  | _ => constr:(InitialRing.NotConstant)
  end.

Add Field RField : Rfield
   (completeness Zeq_bool_complete, constants [IZR_tac], power_tac R_power_theory [Rpow_tac]).