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Removing default lmodType on complex, and adding Rcomplex
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@CohenCyril
CohenCyril committed May 14, 2020
1 parent 12a7240 commit e5727b3
Showing 1 changed file with 53 additions and 51 deletions.
104 changes: 53 additions & 51 deletions theories/complex.v
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Original file line number Diff line number Diff line change
Expand Up @@ -244,6 +244,7 @@ Definition ltc (x y : R[i]) :=
let: a +i* b := x in let: c +i* d := y in
(d == b) && (a < c).

(* TODO: when Pythagorean Fields are added, weaken to Pythagorean Field *)
Definition normc (x : R[i]) : R :=
let: a +i* b := x in sqrtr (a ^+ 2 + b ^+ 2).

Expand Down Expand Up @@ -327,24 +328,27 @@ Definition complex_numMixin := NumMixin lec_normD ltc0_add eq0_normC
ge0_lec_total normCM lec_def ltc_def.
Canonical complex_porderType := POrderType ring_display R[i] complex_numMixin.
Canonical complex_numDomainType := NumDomainType R[i] complex_numMixin.
Canonical complex_numFieldType := [numFieldType of R[i]].
Canonical complex_normedZmodType :=
NormedZmodType R[i] R[i] complex_numMixin.

End ComplexField.
End ComplexField.

Canonical ComplexField.complex_zmodType.
Canonical ComplexField.complex_lmodType.
(* we do not export the canonical structure of lmodType on purpose *)
(* i.e. no: Canonical ComplexField.complex_lmodType. *)
(* indeed, this would prevent C fril having a normed module over C *)
Canonical ComplexField.complex_ringType.
Canonical ComplexField.complex_comRingType.
Canonical ComplexField.complex_unitRingType.
Canonical ComplexField.complex_comUnitRingType.
Canonical ComplexField.complex_idomainType.
Canonical ComplexField.complex_fieldType.
Canonical ComplexField.complex_porderType.
Canonical ComplexField.complex_numDomainType.
Canonical ComplexField.complex_normedZmodType.
Canonical complex_numFieldType (R : rcfType) := [numFieldType of complex R].
Canonical ComplexField.complex_numDomainType.
Canonical ComplexField.complex_numFieldType.
Canonical ComplexField.real_complex_rmorphism.
Canonical ComplexField.real_complex_additive.
Canonical ComplexField.Re_additive.
Expand Down Expand Up @@ -373,49 +377,46 @@ Ltac simpc := do ?
Section ComplexTheory.

Variable R : rcfType.
Implicit Types (k : R) (x y z : R[i]).

Lemma ReiNIm : forall x : R[i], Re (x * 'i%C) = - Im x.
Lemma ReiNIm : forall x, Re (x * 'i%C) = - Im x.
Proof. by case=> a b; simpc. Qed.

Lemma ImiRe : forall x : R[i], Im (x * 'i%C) = Re x.
Lemma ImiRe : forall x, Im (x * 'i%C) = Re x.
Proof. by case=> a b; simpc. Qed.

Lemma complexE x : x = (Re x)%:C + 'i%C * (Im x)%:C :> R[i].
Proof. by case: x => *; simpc. Qed.

Lemma real_complexE x : x%:C = x +i* 0 :> R[i]. Proof. done. Qed.
Lemma real_complexE k : k%:C = k +i* 0 :> R[i]. Proof. done. Qed.

Lemma sqr_i : 'i%C ^+ 2 = -1 :> R[i].
Proof. by rewrite exprS; simpc; rewrite -real_complexE rmorphN. Qed.

Lemma complexI : injective (real_complex R). Proof. by move=> x y []. Qed.

Lemma ler0c (x : R) : (0 <= x%:C) = (0 <= x). Proof. by simpc. Qed.
Lemma ler0c k : (0 <= k%:C) = (0 <= k). Proof. by simpc. Qed.

Lemma lecE : forall x y : R[i], (x <= y) = (Im y == Im x) && (Re x <= Re y).
Lemma lecE : forall x y, (x <= y) = (Im y == Im x) && (Re x <= Re y).
Proof. by move=> [a b] [c d]. Qed.

Lemma ltcE : forall x y : R[i], (x < y) = (Im y == Im x) && (Re x < Re y).
Lemma ltcE : forall x y, (x < y) = (Im y == Im x) && (Re x < Re y).
Proof. by move=> [a b] [c d]. Qed.

Lemma lecR : forall x y : R, (x%:C <= y%:C) = (x <= y).
Proof. by move=> x y; simpc. Qed.
Lemma lecR : forall k k', (k%:C <= k'%:C) = (k <= k').
Proof. by move=> k k'; simpc. Qed.

Lemma ltcR : forall x y : R, (x%:C < y%:C) = (x < y).
Proof. by move=> x y; simpc. Qed.
Lemma ltcR : forall k k', (k%:C < k'%:C) = (k < k').
Proof. by move=> k k'; simpc. Qed.

Lemma conjc_is_rmorphism : rmorphism (@conjc R).
Proof.
split=> [[a b] [c d]|] /=; first by simpc; rewrite [d - _]addrC.
by split=> [[a b] [c d]|] /=; simpc.
Qed.

Lemma conjc_is_scalable : scalable (@conjc R).
Proof. by move=> a [b c]; simpc. Qed.

Canonical conjc_rmorphism := RMorphism conjc_is_rmorphism.
Canonical conjc_additive := Additive conjc_is_rmorphism.
Canonical conjc_linear := AddLinear conjc_is_scalable.

Lemma conjcK : involutive (@conjc R).
Proof. by move=> [a b] /=; rewrite opprK. Qed.
Expand Down Expand Up @@ -448,7 +449,7 @@ Proof. by move: x=> [a b] /=; simpc => /andP [/eqP]. Qed.
(* Todo : extend theory of : *)
(* - signed exponents *)

Lemma conj_ge0 : forall x : R[i], (0 <= x ^*) = (0 <= x).
Lemma conj_ge0 : forall x, (0 <= x ^*) = (0 <= x).
Proof. by move=> [a b] /=; simpc; rewrite oppr_eq0. Qed.

Lemma conjc_nat : forall n, (n%:R : R[i])^* = n%:R.
Expand All @@ -460,10 +461,10 @@ Proof. exact: (conjc_nat 0). Qed.
Lemma conjc1 : (1 : R[i]) ^* = 1.
Proof. exact: (conjc_nat 1). Qed.

Lemma conjc_eq0 : forall x : R[i], (x ^* == 0) = (x == 0).
Lemma conjc_eq0 : forall x, (x ^* == 0) = (x == 0).
Proof. by move=> [a b]; rewrite !eq_complex /= eqr_oppLR oppr0. Qed.

Lemma conjc_inv: forall x : R[i], (x^-1)^* = (x^*%C )^-1.
Lemma conjc_inv: forall x, (x^-1)^* = (x^*%C )^-1.
Proof. exact: fmorphV. Qed.

Lemma complex_root_conj (p : {poly R[i]}) (x : R[i]) :
Expand All @@ -474,7 +475,7 @@ Lemma complex_algebraic_trans (T : comRingType) (toR : {rmorphism T -> R}) :
integralRange toR -> integralRange (real_complex R \o toR).
Proof.
set f := _ \o _ => R_integral [a b].
have integral_real x : integralOver f (x%:C) by apply: integral_rmorph.
have integral_real k : integralOver f (k%:C) by apply: integral_rmorph.
rewrite [_ +i* _]complexE.
apply: integral_add => //; apply: integral_mul => //=.
exists ('X^2 + 1).
Expand Down Expand Up @@ -503,20 +504,48 @@ rewrite realE; simpc; rewrite [0 == _]eq_sym.
by have [] := ltrgtP 0 a; rewrite ?(andbF, andbT, orbF, orbb).
Qed.

Lemma complex_realP (x : R[i]) : reflect (exists y, x = y%:C) (x \is Num.real).
Lemma complex_realP x : reflect (exists k, x = k%:C) (x \is Num.real).
Proof.
case: x=> [a b] /=; rewrite complex_real.
by apply: (iffP eqP) => [->|[c []//]]; exists a.
Qed.

Lemma RRe_real (x : R[i]) : x \is Num.real -> (Re x)%:C = x.
Lemma RRe_real x : x \is Num.real -> (Re x)%:C = x.
Proof. by move=> /complex_realP [y ->]. Qed.

Lemma RIm_real (x : R[i]) : x \is Num.real -> (Im x)%:C = 0.
Lemma RIm_real x : x \is Num.real -> (Im x)%:C = 0.
Proof. by move=> /complex_realP [y ->]. Qed.

End ComplexTheory.

Definition Rcomplex := complex.
Canonical Rcomplex_eqType (R : eqType) := [eqType of Rcomplex R].
Canonical Rcomplex_countType (R : countType) := [countType of Rcomplex R].
Canonical Rcomplex_choiceType (R : choiceType) := [choiceType of Rcomplex R].
Canonical Rcomplex_zmodType (R : rcfType) := [zmodType of Rcomplex R].
Canonical Rcomplex_ringType (R : rcfType) := [ringType of Rcomplex R].
Canonical Rcomplex_comRingType (R : rcfType) := [comRingType of Rcomplex R].
Canonical Rcomplex_unitRingType (R : rcfType) := [unitRingType of Rcomplex R].
Canonical Rcomplex_comUnitRingType (R : rcfType) := [comUnitRingType of Rcomplex R].
Canonical Rcomplex_idomainType (R : rcfType) := [idomainType of Rcomplex R].
Canonical Rcomplex_fieldType (R : rcfType) := [fieldType of Rcomplex R].
Canonical Rcomplex_lmodType (R : rcfType) :=
LmodType R (Rcomplex R) (ComplexField.complex_lmodMixin R).

Module RComplexLMod.
Section RComplexLMod.
Variable R : rcfType.
Implicit Types (k : R) (x y z : Rcomplex R).
Canonical ComplexField.complex_lmodType.

Lemma conjc_is_scalable : scalable (conjc : Rcomplex R -> Rcomplex R).
Proof. by move=> a [b c]; simpc. Qed.
Canonical conjc_linear := AddLinear conjc_is_scalable.

End RComplexLMod.
End RComplexLMod.
Canonical RComplexLMod.conjc_linear.

(* Section RcfDef. *)

(* Variable R : realFieldType. *)
Expand Down Expand Up @@ -1182,33 +1211,6 @@ End Paper_HarmDerksen.

End ComplexClosed.

(* End ComplexInternal. *)

(* Canonical ComplexInternal.complex_eqType. *)
(* Canonical ComplexInternal.complex_choiceType. *)
(* Canonical ComplexInternal.complex_countType. *)
(* Canonical ComplexInternal.complex_ZmodType. *)
(* Canonical ComplexInternal.complex_Ring. *)
(* Canonical ComplexInternal.complex_comRing. *)
(* Canonical ComplexInternal.complex_unitRing. *)
(* Canonical ComplexInternal.complex_comUnitRing. *)
(* Canonical ComplexInternal.complex_iDomain. *)
(* Canonical ComplexInternal.complex_fieldType. *)
(* Canonical ComplexInternal.ComplexField.real_complex_rmorphism. *)
(* Canonical ComplexInternal.ComplexField.real_complex_additive. *)
(* Canonical ComplexInternal.ComplexField.Re_additive. *)
(* Canonical ComplexInternal.ComplexField.Im_additive. *)
(* Canonical ComplexInternal.complex_numDomainType. *)
(* Canonical ComplexInternal.complex_normedZmodType. *)
(* Canonical ComplexInternal.complex_numFieldType. *)
(* Canonical ComplexInternal.conjc_rmorphism. *)
(* Canonical ComplexInternal.conjc_additive. *)
(* Canonical ComplexInternal.complex_decField. *)
(* Canonical ComplexInternal.complex_closedField. *)
(* Canonical ComplexInternal.complex_numClosedFieldType. *)

(* Definition complex_algebraic_trans := ComplexInternal.complex_algebraic_trans. *)

Section ComplexClosedTheory.

Variable R : rcfType.
Expand Down

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