Section ClosedFieldTheory.
Variable C : numClosedFieldType.
Implicit Types a x y z : C.
Definition normCK x : `|x| ^+ 2 = x * x^*.
Let Re2 z := z + z^*.
Definition nnegIm z := (0 <= imaginaryC * (z^* - z)).
Definition argCle y z := nnegIm z ==> nnegIm y && (Re2 z <= Re2 y).
Variant rootC_spec n (x : C) : Type :=
RootCspec (y : C) of if (n > 0)%N then y ^+ n = x else y = 0
& forall z, (n > 0)%N -> z ^+ n = x -> argCle y z.
Fact rootC_subproof n x : rootC_spec n x.
Definition nthroot n x := let: RootCspec y _ _ := rootC_subproof n x in y.
Notation "n .-root" := (nthroot n) (at level 2, format "n .-root") : ring_scope.
Notation "n .-root" := (nthroot n) (only parsing) : ring_scope.
Notation sqrtC := 2.-root.
Definition Re x := (x + x^*) / 2%:R.
Definition Im x := 'i * (x^* - x) / 2%:R.
Notation "'Re z" := (Re z) (at level 10, z at level 8) : ring_scope.
Notation "'Im z" := (Im z) (at level 10, z at level 8) : ring_scope.
Let nz2 : 2%:R != 0 :> C. Proof. by rewrite pnatr_eq0. Qed.
(* Real number subset. *)
Hint Resolve Creal_Re Creal_Im : core.
Fact Re_is_additive : additive Re.
Canonical Im_additive := Additive Im_is_additive.
(* Nth roots. *)
Let argCleP y z :
reflect (0 <= 'Im z -> 0 <= 'Im y /\ 'Re z <= 'Re y) (argCle y z).
(* case Du: sqrCi => [u u2N1] /=. *)
(* have/eqP := u2N1; rewrite -sqrCi eqf_sqr => /pred2P[] //. *)
(* have:= conjCi; rewrite /'i; case_rootC => /= v v2n1 min_v conj_v Duv. *)
(* have{min_v} /idPn[] := min_v u isT u2N1; rewrite negb_imply /nnegIm Du /= Duv. *)
(* rewrite rmorphN conj_v opprK -opprD mulrNN mulNr -mulr2n mulrnAr -expr2 v2n1. *)
(* by rewrite mulNrn opprK ler0n oppr_ge0 (ler_nat _ 2 0). *)
Let neg_unity_root n : (n > 1)%N -> exists2 w : C, w ^+ n = 1 & 'Re w < 0.
(* More properties of n.-root will be established in cyclotomic.v. *)
(* The proper form of the Arithmetic - Geometric Mean inequality. *)
(* Square root. *)

Lemma normC_sum_upper (I : finType) (P : pred I) (F G : I -> C) :
(forall i, P i -> `|F i| <= G i) ->
\sum_(i | P i) F i = \sum_(i | P i) G i ->
forall i, P i -> F i = G i.