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gauss_int.v
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From HB Require Import structures.
From mathcomp Require Import all_ssreflect all_algebra all_field.
From mathcomp Require Import archimedean.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import Order.TTheory GRing.Theory Num.Theory UnityRootTheory.
Open Scope ring_scope.
(** Starting from cyril exercise *)
Section PreliminaryLemmas.
(**
* Preliminaries
Let's extend the library on rings and algebraic numbers
with some easy lemmas first.
** Question -2: prove that if a product of natural numbers is 1 then each factor is 1.
Note that we do not consider nat but the copy of nat which is embeded
in the algebraic numbers algC. The theorem already exists for nat, and
we suggest you use a compatibility lemma numbers between nat and Cnat
*)
Lemma Cnat_mul_eq1 :
{in Num.nat &, forall x y : algC, (x * y == 1) = (x == 1) && (y == 1)}.
Proof.
by move=> x y /natrP [n ->] /natrP [m ->]; rewrite -natrM !pnatr_eq1 muln_eq1.
Qed.
Lemma Cnat_add_eq1 : {in Num.nat &, forall x y : algC,
(x + y == 1) = ((x == 1) && (y == 0)) || ((x == 0) && (y == 1))}.
Proof.
move=> x y /natrP [n ->] /natrP [m ->]; rewrite -natrD !pnatr_eq1 ?pnatr_eq0.
by move: n m => [|[|?]] [|[|?]].
Qed.
(**
** Question -1: The real part of product
*)
Lemma algReM (x y : algC) : 'Re (x * y) = 'Re x * 'Re y - 'Im x * 'Im y.
Proof.
rewrite {1}[x]algCrect {1}[y]algCrect mulC_rect Re_rect //;
by rewrite rpredD ?rpredN // rpredM // ?Creal_Re ?Creal_Im.
Qed.
(**
** Question 0: The imaginary part of product
*)
Lemma algImM (x y : algC) : 'Im (x * y) = 'Re x * 'Im y + 'Re y * 'Im x.
Proof.
rewrite {1}[x]algCrect {1}[y]algCrect mulC_rect Im_rect //;
by rewrite rpredD ?rpredN // rpredM // ?Creal_Re ?Creal_Im.
Qed.
Lemma algReV (x : algC) : 'Re (x^-1) = 'Re x / `|x| ^+ 2.
Proof.
have [|/mulfV H] := boolP (x^* == 0).
rewrite conjC_eq0 => /eqP->.
by rewrite invr0 normr0 (Creal_ReP _ _) ?mul0r.
rewrite -{1}[_ ^-1]mul1r -H -mulrA -invfM.
rewrite {1}[x]algCrect conjC_rect ?Creal_Re ?Creal_Im //.
have F : (x^* * x)^-1 \is Creal.
by rewrite rpredV CrealE rmorphM conjCK mulrC.
rewrite mulrBl -mulrN -['i * _ * _]mulrA Re_rect ?normCKC //.
by rewrite rpredM ?Creal_Re.
by rewrite mulrN rpredN rpredM // Creal_Im.
Qed.
Lemma algImV (x : algC) : 'Im (x^-1) = - ('Im x / `|x| ^+ 2).
Proof.
have [|/mulfV H] := boolP (x^* == 0).
rewrite conjC_eq0 => /eqP->.
by rewrite invr0 normr0 (Creal_ImP _ _) ?mul0r ?oppr0.
rewrite -{1}[_ ^-1]mul1r -H -mulrA -invfM.
rewrite {1}[x]algCrect conjC_rect ?Creal_Re ?Creal_Im //.
have F : (x^* * x)^-1 \is Creal.
by rewrite rpredV CrealE rmorphM conjCK mulrC.
rewrite mulrBl -mulrN -['i * _ * _]mulrA Im_rect ?normCKC //.
- by rewrite mulrN.
- by rewrite rpredM ?Creal_Re.
by rewrite mulrN rpredN rpredM // Creal_Im.
Qed.
Lemma algRe_div (x y : algC) :
'Re (x / y) = ('Re x * 'Re y + 'Im x * 'Im y) / `|y| ^+ 2.
Proof.
by rewrite algReM algReV algImV mulrN opprK !mulrA mulrDl.
Qed.
Lemma algIm_div (x y : algC) :
'Im (x / y) = ('Re y * 'Im x - 'Re x * 'Im y) / `|y| ^+ 2.
Proof.
rewrite algImM algReV algImV addrC mulrN [_ * (_ / _)]mulrA mulrBl.
by rewrite mulrAC.
Qed.
Definition cdivz (x y : int) : int :=
(let q := (x %/ y) in
if (y == 0) || (2%:R * (x %% y)%Z <= `|y|) then
q else q + (-1) ^+ (y < 0)%R)%Z.
Infix "%c/" := cdivz (at level 40) : int_scope.
Lemma cdivz0 x : (x %c/ 0)%Z = 0.
Proof. by rewrite /cdivz eqxx divz0. Qed.
Lemma cdiv0z y : (0 %c/ y)%Z = 0.
Proof. by rewrite /cdivz div0z mod0z mulr0 normr_ge0 orbT. Qed.
Lemma cdivz1 x : (x %c/ 1)%Z = x.
Proof. by rewrite /cdivz oner_eq0 divz1 modz1 normr1 mulr0. Qed.
Lemma cdivzz x : x != 0 -> (x %c/ x)%Z = 1.
Proof.
move=> xNz.
by rewrite /cdivz (negPf xNz) divzz xNz modzz mulr0.
Qed.
Definition cmodz (x y : int) : int := x - (x %c/ y)%Z * y.
Infix "%c%" := cmodz (at level 40) : int_scope.
Lemma cdivz_eq (x y : int) : x = (x %c/ y)%Z * y + (x %c% y)%Z.
Proof. by rewrite /cmodz addrC subrK. Qed.
Lemma cmodz0 x : (x %c% 0)%Z = x.
Proof. by rewrite {2}(cdivz_eq x 0) mulr0 add0r. Qed.
Lemma cmod0z y : (0 %c% y)%Z = 0.
Proof. by rewrite {2}(cdivz_eq 0 y) cdiv0z mul0r add0r. Qed.
Lemma cmodz1 x : (x %c% 1)%Z = 0.
Proof. by rewrite /cmodz cdivz1 mulr1 subrr. Qed.
Lemma cmodzz x : (x %c% x)%Z = 0.
Proof.
have [/eqP->|xNz] := boolP (x == 0); first by rewrite cmod0z.
by rewrite /cmodz cdivzz // mul1r subrr.
Qed.
Lemma cmodz_lt (x y : int) : y != 0 -> (2%:R * `|x %c% y| <= `|y|)%Z.
Proof.
move=> yNz; rewrite /cmodz /cdivz (negPf yNz) /=.
have [mLe|eLm] := boolP (2%:R * (_ %% _)%Z <= `|_|).
rewrite {1}(divz_eq x y) [_ * _ + _]addrC addrK.
by rewrite [`|(_ %% _)%Z|]ger0_norm // modz_ge0.
rewrite [_ * y]mulrDl opprD addrA {1}(divz_eq x y) [_ * _ + _]addrC addrK.
have F := ltz_mod x yNz.
rewrite -normrEsign ler0_norm; last first.
by rewrite subr_le0; apply: ltW.
rewrite mulrN mulrBr opprB lterBDl (_ : 2%:R = 1 + 1) //.
by rewrite mulrDl mul1r lerD // ltW // ltNge.
Qed.
End PreliminaryLemmas.
(**
----
* The ring of Gauss integers
- Ref: exercices de mathematiques oraux X-ENS algebre 1
- Exercice 3.10. ENS Lyon
*)
Section gaussInts.
(**
First we define a predicate for the algebraic numbers which are gauss integers.
*)
Definition gaussInt_subdef : pred algC :=
[pred x | ('Re x \is a Num.int) && ('Im x \is a Num.int)].
Definition gaussInt := [qualify a x | gaussInt_subdef x].
Lemma gaussIntE x :
x \is a gaussInt = (('Re x \is a Num.int) && ('Im x \is a Num.int)).
Proof. by []. Qed.
(**
** Question 1: Prove that integers are gauss integers
*)
Lemma Cint_GI (x : algC) : x \is a Num.int -> x \is a gaussInt.
Proof.
move=> x_int.
rewrite !gaussIntE (Creal_ReP _ _) ?(Creal_ImP _ _) ?Rreal_int //.
by rewrite x_int rpred0.
Qed.
(**
** Question 2: Prove that gauss integers form a subfield
*)
Lemma GI_subring : subring_closed gaussInt.
Proof.
split => [|x y /andP[??] /andP[??]|x y /andP[??] /andP[??]].
- by rewrite Cint_GI.
- by rewrite gaussIntE !raddfB /= ?rpredB.
by rewrite gaussIntE algReM algImM rpredB ?rpredD // rpredM.
Qed.
HB.instance Definition _ :=
GRing.isSubringClosed.Build algC gaussInt_subdef GI_subring.
(**
Finally, we define the type of Gauss Integer, as a sigma type of
algebraic numbers. We soon prove that this is in fact a sub type.
*)
Record GI := GIof {
algGI : algC;
algGIP : algGI \is a gaussInt }.
(** We make the defining property of GI a Hint *)
Hint Resolve algGIP : core.
(**
We provide the subtype property.
- This makes it possible to use the generic operator "val" to get an
algC from a Gauss Integer.
*)
#[verbose]
HB.instance Definition _ := [isSub for algGI].
(**
We deduce that the real and imaginary parts of a GI are integers
*)
Lemma GIRe (x : GI) : 'Re (val x) \is a Num.int.
Proof. by have /andP [] := algGIP x. Qed.
Lemma GIIm (x : GI) : 'Im (val x) \is a Num.int.
Proof. by have /andP [] := algGIP x. Qed.
Hint Resolve GIRe GIIm : core.
Canonical ReGI x := GIof (Cint_GI (GIRe x)).
Definition ImGI x := GIof (Cint_GI (GIIm x)).
(**
We provide a ring structure to the type GI, using the subring
canonical property for the predicate gaussInt
*)
HB.howto GI subComRingType.
HB.instance Definition _ := [Countable of GI by <:].
HB.instance Definition _ := [SubChoice_isSubComRing of GI by <:].
(**
- Now we build the unitRing and comUnitRing structure of gauss
integers. Contrarily to the previous structures, the operator is
not the same as on algebraics. Indeed the invertible algebraics are
not necessarily invertible gauss integers.
- Hence, we define the inverse of gauss integers as follow : if the
algebraic inverse happens to be a gauss integer we recover the
proof and package it together with the element and get a gauss
integer, otherwise, we default to the identity.
- A gauss integer is invertible if the algbraic inverse is a gauss
integer.
*)
Definition invGI (x : GI) := insubd x (val x)^-1.
Definition unitGI (x : GI) := (x != 0) && ((val x)^-1 \is a gaussInt).
(**
** Question 3: prove a few facts in order to find a comUnitRingMixin
for GI, and then instantiate the interfaces of unitRingType and
comUnitRingType.
*)
Fact mulGIr : {in unitGI, left_inverse 1 invGI *%R}.
Proof.
move=> x /andP [x_neq0 xVGI]; rewrite /invGI.
by apply: val_inj; rewrite /= insubdK // mulVr ?unitfE.
Qed.
Fact unitGIP (x y : GI) : y * x = 1 -> unitGI x.
Proof.
rewrite /unitGI => /(congr1 val) /=.
have [-> /eqP|x_neq0] := altP (x =P 0); first by rewrite mulr0 eq_sym oner_eq0.
by move=> /(canRL (mulfK x_neq0)); rewrite mul1r => <- /=.
Qed.
Fact unitGI_out : {in [predC unitGI], invGI =1 id}.
Proof.
move=> x; rewrite inE /= -topredE /= /unitGI.
rewrite negb_and negbK => /predU1P [->|/negPf xGIF];
by apply: val_inj; rewrite /invGI ?val_insubd /= ?xGIF // invr0 if_same.
Qed.
HB.instance Definition _ :=
GRing.ComRing_hasMulInverse.Build GI mulGIr unitGIP unitGI_out.
(**
** Question 4: Show that gauss integers are stable by conjugation.
*)
Lemma conjGIE x : (x^* \is a gaussInt) = (x \is a gaussInt).
Proof.
by rewrite gaussIntE Re_conj Im_conj rpredN.
Qed.
(**
We use this fact to build the conjugation of a gauss Integers
*)
Fact conjGI_subproof (x : GI) : (val x)^* \is a gaussInt.
Proof. by rewrite conjGIE. Qed.
Canonical conjGI x := GIof (conjGI_subproof x).
Fact conjGI_sub : {morph conjGI : a b / a - b}.
Proof. by move=> a b; apply/val_eqP; rewrite /= raddfB. Qed.
HB.instance Definition _ := GRing.isAdditive.Build GI GI conjGI conjGI_sub.
Fact conjGI_multiplicative : multiplicative conjGI.
Proof. by split=> [a b|]; apply/val_eqP; rewrite /= ?(rmorphM,rmorph1). Qed.
HB.instance Definition _ :=
GRing.isMultiplicative.Build GI GI conjGI conjGI_multiplicative.
Lemma algGI_nat n : algGI n%:R = n%:R.
Proof. by elim: n => //= n IH; rewrite -addn1 !natrD -IH. Qed.
Lemma conjGI_nat n : conjGI n%:R = n%:R.
Proof. by apply/val_eqP; rewrite /= algGI_nat conjC_nat. Qed.
Lemma conjGIK : involutive conjGI.
Proof. by move=> x; apply/val_eqP/eqP; exact: conjCK. Qed.
(**
We now define the norm (stasm) for gauss integer, we don't need to
specialize it to gauss integer so we define it over algebraic numbers
instead.
*)
Definition gaussNorm (x : algC) := x * x^*.
(**
** Question 4: Show that the gaussNorm of x is the square of the complex modulus of x
*)
Lemma gaussNormE x : gaussNorm x = `|x| ^+ 2.
Proof. by rewrite normCK. Qed.
(**
** Question 5: Show that the gaussNorm of an gauss integer is a natural number.
*)
Lemma gaussNormCnat (x : GI) : gaussNorm (val x) \is a Num.nat.
Proof. by rewrite /gaussNorm -normCK normC2_Re_Im rpredD // natr_exp_even. Qed.
Hint Resolve gaussNormCnat : core.
Declare Scope GI_scope.
Delimit Scope GI_scope with GI.
Open Scope GI_scope.
Definition normGI (x : GI) := Num.trunc (gaussNorm (val x)).
Local Notation "'N x" := (normGI x%R) (at level 10) : GI_scope.
(**
** Question 6: Show that gaussNorm is multiplicative (on all algC).
*)
Lemma gaussNorm0 : gaussNorm 0 = 0.
Proof. by rewrite /gaussNorm mul0r. Qed.
Lemma normGI0 : 'N 0 = 0%N.
Proof. by rewrite /normGI gaussNorm0 (natrK 0). Qed.
Lemma gaussNorm1 : gaussNorm 1 = 1.
Proof. by rewrite /gaussNorm rmorph1 mulr1. Qed.
Lemma normGI1 : 'N 1 = 1%N.
Proof. by rewrite /normGI gaussNorm1 (natrK 1). Qed.
Lemma gaussNormM : {morph gaussNorm : x y / x * y}.
Proof. by move=> x y; rewrite /gaussNorm rmorphM mulrACA. Qed.
Lemma normGIM x y : 'N (x * y) = ('N x * 'N y)%N.
Proof. by rewrite /normGI gaussNormM truncM. Qed.
Lemma normGIX x n : 'N (x ^+ n) = ('N x ^ n)%N.
Proof.
elim: n => [|n IH].
by rewrite expr0 normGI1 expn0.
by rewrite exprS normGIM IH expnS.
Qed.
Lemma gaussNorm_eq0 (x : GI) : (gaussNorm (algGI x) == 0) = (x == 0).
Proof. by rewrite gaussNormE sqrf_eq0 normr_eq0. Qed.
Lemma normGI_eq0 (x : GI) : ('N x == 0%N) = (x == 0).
Proof.
have /charf0P<- := Cchar.
by rewrite truncK // gaussNorm_eq0.
Qed.
Lemma normGI_gt0 (x : GI) : ('N x > 0)%N = (x != 0).
Proof. by rewrite ltn_neqAle andbT eq_sym normGI_eq0. Qed.
Lemma normGI_le (x y : GI) : y != 0 -> ('N x <= 'N (x * y))%N.
Proof.
rewrite -!normGI_eq0 normGIM; case: ('N _) => // n _.
by rewrite leq_pmulr.
Qed.
Lemma normGI_nat n : 'N n%:R = (n ^ 2)%N.
Proof.
by rewrite /normGI [val _]algGI_nat gaussNormE normr_nat truncX // natrK.
Qed.
Lemma normGIE (x : GI) : ('N(x) = Num.trunc (`|'Re (val x)|)%R ^ 2 +
Num.trunc (`|'Im (val x)|)%R ^ 2)%N.
Proof.
rewrite /normGI gaussNormE normC2_Re_Im truncD ?natr_exp_even //; last first.
by rewrite qualifE /= natr_ge0 // natr_exp_even.
by rewrite -!truncX ?natr_norm_int // !intr_normK.
Qed.
Lemma truncC_Cint (x : algC) :
x \is a Num.int -> x = (-1) ^+ (x < 0)%R * (Num.trunc `|x|)%:R.
Proof.
by move=> xCint; rewrite {1}[x]intrEsign // truncK // natr_norm_int.
Qed.
Lemma normGI_eq1 (x : GI) : ('N(x) == 1)%N = (val x \in [::1;-1;'i;-'i]).
Proof.
apply/idP/idP; last first.
by rewrite normGIE !inE => /or4P[] /eqP->;
rewrite ?raddfN /= ?(Creal_ReP 1 _) ?(Creal_ImP 1 _) ?Re_i ?Im_i //=
?normrN ?normr1 ?normr0 ?trunc0 ?trunc1.
rewrite [val x]algCrect normGIE.
have /andP[/truncC_Cint {2}-> /truncC_Cint {2}->] := algGIP x.
by case: Num.trunc => [|[|m]] //; case: Num.trunc => [|[|n]] // _;
rewrite !(mulr1, mulr0, add0r, addr0); case: (_ < _)%R;
rewrite ?(expr1, expr0, mulrN, mulr1, inE, eqxx, orbT).
Qed.
(**
** Question 7: Find the invertible elements of GI
- This is question 1 of the CPGE exercice
- Suggested strategy: sketch the proof on a paper first, don't let
Coq divert you from your proofsketch
*)
Lemma unitGIE (x : GI) : (x \in GRing.unit) =
(val x \in 4.-unity_root).
Proof.
have eq_algC (a b : algC) : (a == b) = ('Re a == 'Re b) && ('Im a == 'Im b).
rewrite {1}[a]algCrect {1}[b]algCrect -subr_eq0 opprD addrACA -mulrBr.
rewrite -normr_eq0 -sqrf_eq0 normC2_rect ?rpredB ?Creal_Re ?Creal_Im //.
rewrite paddr_eq0 ?real_exprn_even_ge0 // ?rpredB ?Creal_Re ?Creal_Im //.
by rewrite !expf_eq0 /= !subr_eq0.
have N1Creal : (-1: algC) \is Num.real.
by rewrite rpredN -[_ \in _]/(1 \is Num.real).
have oneE : 1 = 1 + 'i * 0 :> algC by rewrite mulr0 addr0.
have N1E : - 1 = - 1 + 'i * 0 :> algC by rewrite mulr0 addr0.
have iE : 'i = 0 + 'i * 1 :> algC by rewrite mulr1 add0r.
have NiE : - 'i = 0 + 'i * (- 1) :> algC by rewrite mulrN1 add0r.
have onerN1 : (1 == -1 :> algC) = false.
by rewrite -subr_eq0 opprK paddr_eq0 ?oner_eq0 ?ler01.
pose my := @id algC.
transitivity (gaussNorm (val x) == 1).
apply/idP/eqP; last first.
by move=> gNx; apply/unitrPr; exists (conjGI x); apply: val_inj.
move=> x_unit; have /(congr1 (gaussNorm \o val)) /= /eqP := mulrV x_unit.
by rewrite gaussNormM gaussNorm1 Cnat_mul_eq1 //= => /andP [/eqP].
rewrite (@mem_unity_roots _ 4 (map my [:: 1; -1; 'i; -'i])) //; last 2 first.
- rewrite /= !unity_rootE /= [(- 'i) ^+ _]exprNn expr1n -signr_odd ?expr0.
by rewrite -[4%N]/(2 * 2)%N exprM sqrCi -signr_odd ?expr0 mulr1 !eqxx.
- rewrite /= .
rewrite /= !in_cons !in_nil /= !negb_or -!andbA !andbT /=.
rewrite ![_ + 'i * _ == _]eq_algC ?Re_rect ?Im_rect //.
rewrite ![_ == -1]eq_sym ![_ == 1]eq_sym oppr_eq0.
by rewrite eqxx onerN1 oner_eq0.
rewrite gaussNormE [val x]algCrect normC2_rect ?Creal_Re ?Creal_Im //.
rewrite Cnat_add_eq1 ?natr_exp_even ?expf_eq0 //=.
rewrite -intr_normK // -['Im _ ^+ 2]intr_normK //.
rewrite !expr2 !Cnat_mul_eq1 ?andbb ?natr_norm_int //.
rewrite !real_eqr_norml ?Creal_Re ?Creal_Im ?ler01 ?andbT //=.
rewrite !inE .
rewrite ![_ + 'i * _ == _]eq_algC
?Re_rect ?Im_rect // ?Creal_Re ?Creal_Im //.
by rewrite andb_orl andb_orr -orbA.
Qed.
Lemma algC_eqE (x y : algC) : (x == y) = (('Re x == 'Re y) && ('Im x == 'Im y)).
Proof.
apply/eqP/andP=> [->//|[/eqP H1 /eqP H2]].
by rewrite [x]algCrect H1 H2 -algCrect.
Qed.
Lemma normGI_unit (x : GI) : ('N(x) == 1)%N = (x \in GRing.unit).
Proof.
rewrite normGI_eq1 unitGIE.
rewrite (@mem_unity_roots _ 4 (map id [:: 1; -1; 'i; -'i])) //.
rewrite /= !unity_rootE /= [(- 'i) ^+ _]exprNn expr1n -signr_odd ?expr0.
by rewrite -[4%N]/(2 * 2)%N exprM sqrCi -signr_odd ?expr0 mulr1 !eqxx.
rewrite /= !in_cons !in_nil /= !negb_or -!andbA !andbT /= eqr_opp.
rewrite -addr_eq0 (eqC_nat 2 0) andTb.
rewrite algC_eqE (Creal_ImP _ _) // Im_i (eqC_nat 0 1) andbF andTb.
rewrite algC_eqE raddfN (Creal_ReP _ _) //= Re_i oppr0 (eqC_nat 1 0) andFb andTb.
rewrite algC_eqE !raddfN /= (Creal_ImP _ _) // Im_i oppr0 (eqC_nat 0 1) andbF andTb.
by rewrite -addr_eq0 (@mulrn_eq0 _ 'i 2) negb_or neq0Ci.
Qed.
Fact GI_idomainAxiom (x y : GI) : x * y = 0 -> (x == 0) || (y == 0).
Proof.
move=> /(congr1 (gaussNorm \o val)) /= /eqP.
by rewrite gaussNorm0 gaussNormM mulf_eq0 !gaussNorm_eq0.
Qed.
HB.instance Definition _ := GRing.ComUnitRing_isIntegral.Build GI
GI_idomainAxiom.
Fact divGI_subproof (x y : int) : x%:~R + 'i * y%:~R \is a gaussInt.
Proof. by rewrite gaussIntE Re_rect ?Im_rect ?Rreal_int ?intr_int. Qed.
Definition divGI (x y : GI) : GI :=
let zr := Num.floor ('Re (val x) * 'Re (val y) + 'Im (val x) * 'Im (val y)) in
let zi := Num.floor ('Re (val y) * 'Im (val x) - 'Re (val x) * 'Im (val y)) in
let n := 'N y in
GIof (divGI_subproof (cdivz zr n) (cdivz zi n)).
Notation " x %/ y " := (divGI x y) : GI_scope.
Lemma divGI0 x : x %/ 0 = 0.
Proof.
apply/val_eqP=> /=.
by rewrite !raddf0 !(mul0r, mulr0, subrr, add0r, floor0, normGI0, cdiv0z).
Qed.
Lemma div0GI y : 0 %/ y = 0.
Proof.
apply/val_eqP=> /=.
by rewrite !raddf0 !(mul0r, mulr0, subrr, add0r, floor0, normGI0, cdiv0z).
Qed.
Lemma divGI1 x : x %/ 1 = x.
Proof.
apply/val_eqP=> /=.
have /andP[CR CI] := algGIP x.
rewrite normGI1 !cdivz1 (Creal_ReP 1 _) ?(Creal_ImP 1 _) //
!(mul1r, mulr1, mulr0, mul0r, addr0, add0r, subr0).
by rewrite 2?floorK ?realr_int // -algCrect.
Qed.
Lemma divGIxx (x : GI) : x != 0 -> x %/ x = 1.
Proof.
move=> xNz; apply/val_eqP => /=.
rewrite subrr floor0 cdiv0z mulr0 addr0.
have := xNz.
rewrite -normGI_eq0.
rewrite /normGI gaussNormE [val x]algCrect normC2_rect ?(realr_int, intr_int) //.
set u :=_ + _ * _ => uNz.
have->: Num.floor u = Num.trunc u.
apply: floor_def.
rewrite [(_+ 1)%Z]addrC -intS trunc_itv //.
rewrite addr_ge0 // -expr2 real_exprn_even_ge0 ?(realr_int, intr_int) //.
by rewrite cdivzz ?mul1r ?subrr.
Qed.
Definition modGI (x y : GI) : GI := x - (x %/ y)%GI * y.
Notation " x %% y " := (modGI x y) : GI_scope.
Lemma modGI0 x : x %% 0 = x.
Proof. by apply/val_eqP; rewrite /= !raddf0 !mulr0 subr0. Qed.
Lemma mod0GI y : 0 %% y = 0.
Proof.
apply/val_eqP.
by rewrite /= !(raddf0, mul0r, mulr0, add0r, floor0, cdiv0z).
Qed.
Lemma modGI1 x : x %% 1 = 0.
Proof. by rewrite /modGI divGI1 mulr1 subrr. Qed.
Lemma divGI_eq (x y : GI) : x = (x %/ y)%GI * y + (x %% y)%GI.
Proof. by rewrite /modGI addrC subrK. Qed.
Lemma ltn_modGI(x y : GI) : ('N (x %% y)%GI < 'N y)%N = (y != 0).
Proof.
have [/eqP->|yNz] := boolP (y == 0).
by rewrite normGI0 modGI0.
have /ltn_pmul2r<-: (0 < 'N(y) ^ 2)%N by rewrite sqrn_gt0 lt0n normGI_eq0.
rewrite -{1}normGI_nat -!normGIM /modGI /divGI.
set Ux := Num.floor _.
set Uy := Num.floor _.
have UxRe : Ux%:~R = 'Re (algGI x / algGI y * ('N y)%:R).
rewrite algReM ['Re _%:R](Creal_ReP _ _) ?qualifE /= ?ler0n //.
rewrite ?['Im _%:R](Creal_ImP _ _) ?qualifE /= ?ler0n // mulr0 subr0.
rewrite /normGI truncK // algRe_div -gaussNormE divfK; last first.
by rewrite gaussNorm_eq0.
by rewrite floorK // rpredD // rpredM.
have UyIm : Uy%:~R = 'Im (algGI x / algGI y * ('N(y))%GI%:R).
rewrite algImM ['Re _%:R](Creal_ReP _ _) ?qualifE /= ?ler0n //.
rewrite ?['Im _%:R](Creal_ImP _ _) ?qualifE /= ?ler0n // mulr0 add0r mulrC.
rewrite /normGI truncK // algIm_div -gaussNormE divfK; last first.
by rewrite gaussNorm_eq0.
by rewrite floorK // rpredB // rpredM.
rewrite ['N (_ * _)]/normGI /= -[algGI x](divfK yNz).
rewrite -mulrBl -mulrA -[algGI y * _]mulrC mulrA algGI_nat mulrBl.
rewrite [_ * _%:R]algCrect -UxRe -UyIm [_ * _%:R]mulrDl -['i * _ * _]mulrA.
rewrite {1}(cdivz_eq Ux ('N y)) {1}(cdivz_eq Uy ('N y)).
rewrite subC_rect ![_ + cmodz _ _]addrC.
rewrite rmorphD /= rmorphM /= addrK.
rewrite [(_ + _)%:~R]rmorphD /= rmorphM /= addrK.
rewrite !gaussNormM gaussNormE normC2_rect ?(Rreal_int, intr_int) //.
rewrite truncM //; last by rewrite rpredD // natr_exp_even // intr_int.
rewrite mulnC ltn_pmul2l; last by rewrite lt0n normGI_eq0.
rewrite -!rmorphXn /= -!rmorphD /=.
rewrite -[_ + _]gez0_abs ?natrK; last first.
by rewrite addr_ge0 // exprn_even_ge0.
set x1 := _ ^+ 2; set x2 := _ ^+ 2.
apply: leq_ltn_trans (_ : (`|x1| + `|x2| < _)%N).
have := leqD_dist (x1) (x1 - x2) (-x2).
by rewrite !opprK subrK opprB [_ + (_ - _)]addrC subrK [(`|_| + _)%N]addnC.
rewrite -(ltn_pmul2l (isT: (0 < 2 ^ 2)%N)) mulnDr.
apply: leq_ltn_trans (_ : 2 * 'N y ^ 2 < _)%N; last first.
by rewrite ltn_mul2r sqrn_gt0 lt0n normGI_eq0 yNz.
have F : Posz ('N y) != 0.
by rewrite eqz_nat normGI_eq0.
by rewrite mul2n -addnn leq_add // !abszX -!expnMn leq_exp2r //
-(@ler_nat (Num.NumDomain.clone _ int)) natrM !natz /= cmodz_lt.
Qed.
Lemma ltn_modGIN0 x y : y != 0 -> ('N (x %% y)%GI < 'N y)%N.
Proof. by rewrite ltn_modGI. Qed.
Lemma modGIxx x : (x %% x)%GI = 0.
Proof.
have [/eqP->|xNz] := boolP (x == 0); first by rewrite mod0GI.
by rewrite /modGI divGIxx // mul1r subrr.
Qed.
Lemma divGIKl (x y : GI) : x != 0 -> (y * x %/ x) = y.
Proof.
move=> xNz.
apply/eqP; rewrite eq_sym -subr_eq0.
have := xNz; rewrite -(ltn_modGI (y * x)).
have -> : ((y * x) %% x)%GI = (y - ((y * x) %/ x)%GI) * x.
by rewrite mulrBl {2}(divGI_eq (y * x) x) [_ + (_ %% _)%GI]addrC addrK.
by rewrite normGIM -{2}['N x]mul1n ltn_mul2r ltnS leqn0 normGI_eq0 => /andP[].
Qed.
Lemma divGIKr (x y : GI) : x != 0 -> (x * y %/ x) = y.
Proof. by rewrite mulrC; exact: divGIKl. Qed.
Lemma modGIKl (x y : GI) : (y * x %% x) = 0.
Proof.
have [/eqP->|xNz] := boolP (x == 0); first by rewrite modGI0 mulr0.
by rewrite /modGI divGIKl // subrr.
Qed.
Lemma modGIKr (x y : GI) : (x * y %% x) = 0.
Proof. by rewrite mulrC modGIKl. Qed.
Definition dvdGI x y := (y %% x)%GI == 0.
Notation " x %| y " := (dvdGI x y) : GI_scope.
Lemma dvdGI0 x : (x %| 0)%GI.
Proof. by rewrite /dvdGI mod0GI. Qed.
Lemma dvdGIP (x y : GI) :
reflect (exists q : GI, y = q * x) (x %| y)%GI.
Proof.
apply: (iffP idP) => [/eqP xDy|[q ->]].
exists (y %/ x)%GI; first by rewrite {1}(divGI_eq y x) xDy addr0.
by rewrite /dvdGI modGIKl.
Qed.
Lemma dvd0GI x : (0 %| x) = (x == 0).
Proof.
apply/dvdGIP/eqP => [[q ->]|->]; first by rewrite mulr0.
by exists 0; rewrite mulr0.
Qed.
Lemma dvdGI_mull z x y : (x %| y) -> (x %| z * y).
Proof. by move=> /dvdGIP[u ->]; apply/dvdGIP; exists (z * u); exact: mulrA. Qed.
Lemma dvdGI_mulr z x y : (x %| y) -> (x %| y * z).
Proof. by rewrite mulrC; exact: dvdGI_mull. Qed.
Lemma dvdGIxx x : x %| x.
Proof. by rewrite /dvdGI modGIxx. Qed.
Lemma dvdGI_norm x y : x %| y -> ('N x %| 'N y)%N.
Proof. by move=> /dvdGIP[z ->]; rewrite normGIM dvdn_mull // dvdnn. Qed.
Lemma dvd1GI x : (1 %| x) .
Proof. by apply/dvdGIP; exists x; rewrite mulr1. Qed.
Lemma dvdGI1 x : (x %| 1) = ('N x == 1%N).
Proof.
apply/idP/idP => [H|].
by have := dvdGI_norm H; rewrite normGI1 dvdn1.
rewrite normGI_unit => H; apply/dvdGIP; exists x^-1.
by apply/eqP; rewrite mulrC eq_sym -unitrE.
Qed.
Lemma divGIK (x y : GI) : x %| y -> (y %/ x)%GI * x = y.
Proof.
have [/eqP->|nZx /dvdGIP[q ->]] := boolP (x == 0).
by rewrite dvd0GI mulr0 => /eqP->.
by rewrite divGIKl.
Qed.
Lemma dvdGI_add x y z: (x %| y) -> (x %| z) -> (x %| y + z).
Proof.
move=> /dvdGIP[q1->] /dvdGIP[q2->]; apply/dvdGIP; exists (q1 + q2).
by rewrite mulrDl.
Qed.
Lemma dvdGI_nat_dvdz_Re n x :
n%:R %| x -> (n %| `|Num.floor ('Re (algGI x))|)%N.
Proof.
case/dvdGIP=> q /val_eqP/eqP /(congr1 (fun x => Re x)) /=.
case: x => /= ax; rewrite qualifE => /andP[Rx Ix].
case: q => /= aq; rewrite qualifE => /andP[Rq Iq].
rewrite [aq]algCrect mulrDl algGI_nat -['i * _ * _]mulrA Re_rect; last 2 first.
by rewrite rpredM // Rreal_int // intr_nat.
by rewrite rpredM // Rreal_int // intr_nat.
move=> /(congr1 Num.norm) /eqP.
rewrite [`|_* _%:R|]normrM -{1}(floorK Rx) -{1}(floorK Rq).
rewrite normr_nat -!intr_norm -(intrM _ _ (Posz n)) eqr_int.
rewrite -2![`|_|]abszE -PoszM => /eqP[H].
by apply/dvdnP; exists (`|Num.floor ('Re aq)|)%N.
Qed.
Lemma dvdGI_nat_dvdz_Im n x :
n%:R %| x -> (n %| `|Num.floor ('Im (algGI x))|)%N.
Proof.
case/dvdGIP=> q /val_eqP/eqP/(congr1 (fun x => Im x)) /=.
case: x => /= ax; rewrite qualifE => /andP[Rx Ix].
case: q => /= aq; rewrite qualifE => /andP[Rq Iq].
rewrite [aq]algCrect mulrDl algGI_nat -['i * _ * _]mulrA Im_rect; last 2 first.
by rewrite rpredM // Rreal_int // intr_nat.
by rewrite rpredM // Rreal_int // intr_nat.
move=> /(congr1 Num.norm) /eqP.
rewrite [`|_* _%:R|]normrM -{1}(floorK Ix) -{1}(floorK Iq).
rewrite normr_nat -!intr_norm -(intrM _ _ (Posz n)) eqr_int.
rewrite -2![`|_|]abszE -PoszM => /eqP[H].
by apply/dvdnP; exists (`|Num.floor ('Im aq)|)%N.
Qed.
Lemma conjGI_dvd x y : x %| y -> (conjGI x) %| (conjGI y).
Proof.
case/dvdGIP=> q ->; apply/dvdGIP; exists (conjGI q).
by rewrite rmorphM.
Qed.
Fact iGI_proof : 'i \is a gaussInt.
Proof. by rewrite gaussIntE Re_i Im_i int_num0 int_num1. Qed.
Definition iGI := GIof iGI_proof.
Lemma dvdGI_norm_even x : ~~ odd ('N x) = ((1 + iGI) %| x).
Proof.
apply/idP/idP => [Ex|/dvdGIP[u ->]]; last first.
rewrite normGIM {2}/normGI gaussNormE normC2_Re_Im.
rewrite !raddfD /= Re_i Im_i (Creal_ReP _ _) // (Creal_ImP _ _) //.
by rewrite add0r addr0 expr1n (natrK 2) oddM negb_and orbT.
apply/dvdGIP.
have := algGIP x; rewrite qualifE => / andP[].
have := Ex; rewrite normGIE oddD !oddX /= negb_add.
set m := 'Re _; set n := 'Im _ => /eqP Omn Cm Cn.
suff FF : (n + m)/2%:R + 'i * ((n - m)/2%:R) \is a gaussInt.
exists (GIof FF); apply/val_eqP => /=.
rewrite -{2}['i]mulr1 mulC_rect !mulr1 mul1r -mulrBl -mulrDl.
rewrite opprB [_ + (m - n)]addrC addrA subrK.
rewrite -addrA [_ + (_ - _)]addrC subrK.
have F u : (u * 2%:R = u + u) by rewrite mulrDr mulr1.
by rewrite -!F !mulfK 1?[algGI x]algCrect ?(eqC_nat _ 0).
rewrite gaussIntE Re_rect ?Im_rect; last 4 first.
- by rewrite !(rpredM, rpredV) 1? rpredD ?Rreal_int.
- by rewrite !(rpredM, rpredV) 1? rpredB ?rpredD ?Rreal_int.
- by rewrite !(rpredM, rpredV) 1? rpredD ?Rreal_int.
- by rewrite !(rpredM, rpredV) 1? rpredB ?rpredD ?Rreal_int.
rewrite (intrEsign Cm) (intrEsign Cn).
rewrite -(truncK (natr_norm_int Cm)) -(truncK (natr_norm_int Cn)).
rewrite -[Num.trunc `|m|]odd_double_half -[Num.trunc `|n|]odd_double_half.
rewrite Omn !natrD !mulrDr ![(-1) ^+ _]signrE.
set u := nat_of_bool _; set v := nat_of_bool _; set w := nat_of_bool _.
set x1 := _./2; set y1 := _./2.
have F : 2%:R != 0 :> algC by rewrite (eqC_nat 2 0).
apply/andP; split.
rewrite addrAC addrA -mulrDl -addrA.
rewrite addrAC !addrA -[1 + 1](natrD _ 1 1) addnn.
rewrite -!muln2 !natrM !mul1r [_ * v%:R]mulrC.
rewrite ![v%:R * _]mulrBr !mulrA !(mulrBl, mulrDl) !mulfK //.
by rewrite !mul1r !(rpredB, rpredD, rpredN, rpredM) // intr_nat // natr_nat.
rewrite addrAC opprD addrA -mulrBl opprB [(_ - _) + (_ - _)]addrC !addrA subrK.
rewrite -!muln2 !natrM [_ * v%:R]mulrC.
rewrite ![v%:R * _]mulrBr !mulrA !(mulrBl, mulrDl) !mulfK //.
by rewrite !mul1r !(rpredB, rpredD, rpredN, rpredM) // intr_nat // natr_nat.
Qed.
Fixpoint gcdGI_rec (n : nat) (xx yy : GI) {struct n} :=
let rr := modGI xx yy in
if rr == 0 then yy else
if n is n1.+1 then gcdGI_rec n1 yy rr else rr.
Definition gcdGI x y :=
let: (x1, y1) := if ('N x < 'N y)%N then (y, x) else (x, y) in
if x1 == 0 then y1 else
gcdGI_rec ('N x1) x1 y1.
Lemma gcd0GI : left_id 0 gcdGI.
Proof.
move=> x; rewrite /gcdGI.
have [/eqP->|xNz]:= boolP (x == 0).
by rewrite ltnn eqxx.
rewrite normGI0 normGI_gt0 xNz (negPf xNz).
have : 'N x != 0%N by rewrite normGI_eq0.
by case: ('N _) => [|[|v]]; rewrite //= !(mod0GI, modGI0) (negPf xNz) eqxx.
Qed.
Lemma gcdGI0 : right_id 0 gcdGI.
Proof.
move=> x; rewrite /gcdGI.
have [/eqP->|xNz]:= boolP (x == 0).
by rewrite ltnn eqxx.
rewrite normGI0 /= (negPf xNz).
by case: ('N _) => [|[|v]] //= ; rewrite !(modGI0,mod0GI) (negPf xNz) ?eqxx.
Qed.
Lemma gcdGI_recE m n x y : ('N y <= m)%N -> ('N y <= n)%N
-> ('N y < 'N x)%N -> gcdGI_rec m x y = gcdGI_rec n x y.
Proof.
elim: m n x y => [|m Hrec] [|n] //= x1 y1.
- rewrite leqn0 normGI_eq0 => /eqP=> -> _.
rewrite normGI0 normGI_gt0 modGI0 => /negPf-> /=.
by case: n => [|n]; rewrite /= mod0GI eqxx.
- rewrite leqn0 normGI_eq0 => _ /eqP=> -> _.
rewrite modGI0; case: (boolP (x1 == 0)) => // x1Nz.
by case: m {Hrec} =>[|m]; rewrite /= mod0GI eqxx.
case: ifP => Epq Sm Sn Sq //; rewrite ?Epq //.
case: (eqVneq y1 0) => [->|y1Nz].
by case: n m {Sm Sn Hrec} => [|m] [|n] //=; rewrite mod0GI eqxx.
apply: Hrec; last by rewrite ltn_modGI.
by rewrite -ltnS (leq_trans _ Sm) // ltn_modGI.
by rewrite -ltnS (leq_trans _ Sn) // ltn_modGI.
Qed.
Lemma gcdGIE x y :
gcdGI x y = if ('N x < 'N y)%N
then gcdGI (y %% x) x else gcdGI (x %% y) y.
Proof.
case: (eqVneq x 0) => [-> | xNz].
by rewrite mod0GI modGI0 gcd0GI gcdGI0 if_same.
case: (eqVneq y 0) => [-> | yNz].
by rewrite mod0GI modGI0 gcd0GI gcdGI0 if_same.
rewrite /gcdGI.
case: ltnP; rewrite (negPf xNz, negPf yNz) //=.
move=> ltxy; rewrite ltn_modGI (negPf xNz) //=.
rewrite -(ltn_predK ltxy) /=; case: eqP => [->|].
by case: ('N x) => [|[|s]]; rewrite /= modGI0 (negPf xNz) // mod0GI eqxx.
move/eqP=> yxNz; rewrite (negPf xNz).
apply: gcdGI_recE => //; last by rewrite ltn_modGI.
by rewrite -ltnS (ltn_predK ltxy) (leq_trans _ ltxy) ?leqW // ltn_modGI.
by rewrite ltnW // ltn_modGI.
move=> leyx; rewrite ltn_modGI (negPf yNz) //=.
have x_gt0: ('N x > 0)%N by rewrite normGI_gt0.
rewrite -(prednK x_gt0) /=; case: eqP => [->|].
by case: ('N y)%N => [|[|s]]; rewrite /= modGI0 (negPf yNz) // mod0GI eqxx.
move/eqP=> xyNz; rewrite (negPf yNz).
apply: gcdGI_recE => //; last by rewrite ltn_modGI.
by rewrite -ltnS (prednK x_gt0) (leq_trans _ leyx) // ltn_modGI.
by rewrite ltnW // ltn_modGI.
Qed.
Lemma gcd1GIE y :
gcdGI 1 y = if ('N y == 1)%N then y else 1.
Proof.
rewrite gcdGIE normGI1; case: leqP => [|H].
rewrite leq_eqVlt; case: eqP => [/eqP|/= _].
by rewrite -dvdGI1 => /eqP->; rewrite gcd0GI.
rewrite ltnS leqn0 normGI_eq0 => /eqP->.
by rewrite modGI0 gcdGI0.
rewrite modGI1 gcd0GI.
by move: H; rewrite ltnNge leq_eqVlt negb_or => /andP[/negPf->].
Qed.
Lemma gcd1GI_norm y : 'N(gcdGI 1 y) = 1%N.
Proof. by rewrite gcd1GIE; case: eqP; rewrite ?normGI1. Qed.
Lemma gcdGI1 y : gcdGI y 1 = 1.
Proof.
rewrite gcdGIE normGI1; case: leqP => [_|].
by rewrite modGI1 gcd0GI.
rewrite ltnS leqn0 normGI_eq0 => /eqP->.
by rewrite modGI0 gcdGI0.
Qed.
Lemma gcdGIxx : idempotent_op gcdGI.
Proof. by move=> x; rewrite gcdGIE ltnn modGIxx gcd0GI. Qed.
Lemma dvdGI_mod d x y : d %| x -> (d %| y) = (d %| y %% x).
Proof.
case: (altP (x =P 0)) => [-> | nZx]; first by rewrite modGI0.
case: (altP (d =P 0)) => [-> | nZd /dvdGIP[q1 ->]].
by rewrite dvd0GI => /eqP->; rewrite modGI0.
apply/dvdGIP/dvdGIP=> [] [q2 Hq2].
rewrite /modGI Hq2 !mulrA -mulrBl.
by set u := _ - _; exists u.
rewrite (divGI_eq y (q1 * d)) Hq2 mulrA -mulrDl.
by set u := _ + _; exists u.
Qed.
Lemma dvdGI_gcdlr x y : (gcdGI x y %| x) && (gcdGI x y %| y).
Proof.
elim: {x y}minn {-2}x {-2}y (leqnn (minn ('N y) ('N x))) => [x y|r IH x y].
rewrite geq_min !leqn0 !normGI_eq0.
by case/pred2P=>->;
rewrite (gcd0GI, gcdGI0) dvdGIxx ?andbT dvdGI0.
case: (eqVneq x 0) => [-> _|nZx].
by rewrite gcd0GI dvdGIxx andbT dvdGI0.
case: (eqVneq y 0) => [->|nZy].
by rewrite gcdGI0 dvdGIxx /= dvdGI0.
rewrite gcdGIE minnC /minn; case: ltnP => [lt_xy | le_xy] le_yr.
suffices /IH/andP[E1 E2]: (minn ('N x) ('N (y %% x)%GI) <= r)%N.
by rewrite E2 (dvdGI_mod _ E2).
by rewrite geq_min orbC -ltnS (leq_trans _ le_yr) ?ltn_modGI.
suffices /IH/andP[E1 E2] : (minn ('N y) ('N (x %% y)%GI) <= r)%N.
by rewrite E2 andbT (dvdGI_mod _ E2).
by rewrite geq_min orbC -ltnS (leq_trans _ le_yr) ?ltn_modGI.
Qed.
Lemma dvdGI_gcdl x y : gcdGI x y %| x.
Proof. by case/andP: (dvdGI_gcdlr x y). Qed.
Lemma dvdGI_gcdr x y : gcdGI x y %| y.
Proof. by case/andP: (dvdGI_gcdlr x y). Qed.
Lemma gcdGI_eq0 x y : (gcdGI x y == 0) = ((x == 0) && (y == 0)).
Proof.
have [/eqP->|/eqP nZx] := boolP (x == 0).
by rewrite gcd0GI.
have := dvdGI_gcdl x y.
by case:eqP => //->; rewrite dvd0GI => /eqP.
Qed.
Lemma dvdGI_leq x y : y != 0 -> x %| y -> ('N x <= 'N y)%N.
Proof.
move=> nZy /dvdGIP[q qE]; have := nZy.
rewrite qE -normGI_eq0 normGIM muln_eq0 negb_or => /andP[H1 H2].
by rewrite -[X in (X <= _)%N] mul1n leq_pmul2r lt0n.
Qed.
Lemma leq_gcdGIl (x y : GI) : x != 0 -> ('N (gcdGI x y) <= 'N x)%N.
Proof. by move=> nZx; apply: dvdGI_leq => //; exact: dvdGI_gcdl. Qed.
Lemma leq_gcdGIr (x y : GI) : y != 0 -> ('N (gcdGI x y) <= 'N y)%N.
Proof. by move=> nZy; move: (dvdGI_gcdr x y); apply: dvdGI_leq. Qed.
Lemma dvdGI_trans : transitive dvdGI.
Proof.
move=> x y z /dvdGIP[qx -> /dvdGIP[qy ->]].
by apply/dvdGIP; exists (qy * qx); rewrite mulrA.
Qed.
Lemma dvdGI_gcd x y z : x %| gcdGI y z = (x %| y) && (x %| z).
Proof.
apply/idP/andP=> [dv_xyz | [dv_xy dv_xz]].
by rewrite ?(dvdGI_trans dv_xyz) ?dvdGI_gcdl ?dvdGI_gcdr.
move: (leqnn (minn ('N z) ('N y))) dv_xy dv_xz.
elim: {y z}minn {-2}y {-2}z => [|r IH] y z.
rewrite geq_min !leqn0 !normGI_eq0.
by case/pred2P=> ->; rewrite ?(gcd0GI, gcdGI0).
case: (eqVneq y 0) => [-> _|nZy]; first by rewrite gcd0GI.
case: (eqVneq z 0) => [->|nZz]; first by rewrite gcdGI0.
rewrite gcdGIE minnC /minn; case: ltnP => Czy le_r dv_y dv_z.
apply: IH => //; last by rewrite -(dvdGI_mod _ dv_y).
by rewrite geq_min orbC -ltnS (leq_trans _ le_r) ?ltn_modGI.
apply: IH => //; last by rewrite -(dvdGI_mod _ dv_z).
by rewrite geq_min orbC -ltnS (leq_trans _ le_r) ?ltn_modGI.
Qed.