path: root/mathcomp/algebra/ssrint.v

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrfun ssrbool eqtype ssrnat choice seq.
From mathcomp
Require Import fintype finfun bigop ssralg ssrnum poly.
Import GRing.Theory Num.Theory.

(******************************************************************************)
(* This file develops a basic theory of signed integers, defining:            *)
(*         int == the type of signed integers, with two constructors Posz for *)
(*                non-negative integers and Negz for negative integers. It    *)
(*                supports the realDomainType interface (and its parents).    *)
(*        n%:Z == explicit cast from nat to int (:= Posz n); displayed as n.  *)
(*                However (Posz m = Posz n) is displayed as (m = n :> int)    *)
(*                (and so are ==, != and <>)                                  *)
(*                Lemma NegzE : turns (Negz n) into - n.+1%:Z.                *)
(*      x *~ m == m times x, with m : int;                                    *)
(*                convertible to x *+ n if m is Posz n                        *)
(*                convertible to x *- n.+1 if m is Negz n.                    *)
(*       m%:~R == the image of m : int in a generic ring (:= 1 *~ m).         *)
(*       x ^ m == x to the m, with m : int;                                   *)
(*                convertible to x ^+ n if m is Posz n                        *)
(*                convertible to x ^- n.+1 if m is Negz n.                    *)
(*       sgz x == sign of x : R,                                              *)
(*                equals (0 : int) if and only x == 0,                        *)
(*                equals (1 : int) if x is positive                           *)
(*                and (-1 : int) otherwise.                                   *)
(*      `|m|%N == the n : nat such that `|m|%R = n%:Z, for m : int.           *)
(*  `|m - n|%N == the distance between m and n; the '-' is specialized to     *)
(*                the int type, so m and n can be either of type nat or int   *)
(*                thanks to the Posz coercion; m and n are however parsed in  *)
(*                the %N scope. The IntDist submodule provides this notation  *)
(*                and the corresponding theory independently of the rest of   *)
(*                of the int and ssralg libraries (and notations).            *)
(* Warning: due to the declaration of Posz as a coercion, two terms might be  *)
(* displayed the same while not being convertible, for instance:              *)
(* (Posz (x - y)) and (Posz x) - (Posz y) for x, y : nat.                     *)
(******************************************************************************)


Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Delimit Scope int_scope with Z.
Local Open Scope int_scope.

(* Defining int *)
CoInductive int : Set := Posz of nat | Negz of nat.
(* This must be deferred to module DistInt to work around the design flaws of *)
(* the Coq module system.                                                     *)
(* Coercion Posz : nat >-> int. *)

Notation "n %:Z" := (Posz n)
  (at level 2, left associativity, format "n %:Z", only parsing) : int_scope.
Notation "n %:Z" := (Posz n)
  (at level 2, left associativity, format "n %:Z", only parsing) : ring_scope.

Notation "n = m :> 'in' 't'" := (Posz n = Posz m)
  (at level 70, m at next level, format "n  =  m  :>  'in' 't'") : ring_scope.
Notation "n == m :> 'in' 't'" := (Posz n == Posz m)
  (at level 70, m at next level, format "n  ==  m  :>  'in' 't'") : ring_scope.
Notation "n != m :> 'in' 't'" := (Posz n != Posz m)
  (at level 70, m at next level, format "n  !=  m  :>  'in' 't'") : ring_scope.
Notation "n <> m :> 'in' 't'" := (Posz n <> Posz m)
  (at level 70, m at next level, format "n  <>  m  :>  'in' 't'") : ring_scope.

Definition natsum_of_int (m : int) : nat + nat :=
  match m with Posz p => inl _ p | Negz n => inr _ n end.

Definition int_of_natsum (m : nat + nat) :=
  match m with inl p => Posz p | inr n => Negz n end.

Lemma natsum_of_intK : cancel natsum_of_int int_of_natsum.
Proof. by case. Qed.

Definition int_eqMixin := CanEqMixin natsum_of_intK.
Definition int_countMixin := CanCountMixin natsum_of_intK.
Definition int_choiceMixin := CountChoiceMixin int_countMixin.
Canonical int_eqType := Eval hnf in EqType int int_eqMixin.
Canonical int_choiceType := Eval hnf in ChoiceType int int_choiceMixin.
Canonical int_countType := Eval hnf in CountType int int_countMixin.

Lemma eqz_nat (m n : nat) : (m%:Z == n%:Z) = (m == n). Proof. by []. Qed.

Module intZmod.
Section intZmod.

Definition addz (m n : int) :=
  match m, n with
    | Posz m', Posz n' => Posz (m' + n')
    | Negz m', Negz n' => Negz (m' + n').+1
    | Posz m', Negz n' => if n' < m' then Posz (m' - n'.+1) else Negz (n' - m')
    | Negz n', Posz m' => if n' < m' then Posz (m' - n'.+1) else Negz (n' - m')
  end.

Definition oppz m := nosimpl
  match m with
    | Posz n => if n is (n'.+1)%N then Negz n' else Posz 0
    | Negz n => Posz (n.+1)%N
  end.

Local Notation "0" := (Posz 0) : int_scope.
Local Notation "-%Z" := (@oppz) : int_scope.
Local Notation "- x" := (oppz x) : int_scope.
Local Notation "+%Z" := (@addz) : int_scope.
Local Notation "x + y" := (addz x y) : int_scope.
Local Notation "x - y" := (x + - y) : int_scope.

Lemma PoszD : {morph Posz : m n / (m + n)%N >-> m + n}. Proof. by []. Qed.

Local Coercion Posz : nat >-> int.

Lemma NegzE (n : nat) : Negz n = - n.+1. Proof. by []. Qed.

Lemma int_rect (P : int -> Type) :
  P 0 -> (forall n : nat, P n -> P (n.+1))
  -> (forall n : nat, P (- n) -> P (- (n.+1)))
  -> forall n : int, P n.
Proof.
by move=> P0 hPp hPn []; elim=> [|n ihn]//; do ?[apply: hPn | apply: hPp].
Qed.

Definition int_rec := int_rect.
Definition int_ind := int_rect.

CoInductive int_spec (x : int) : int -> Type :=
| ZintNull of x = 0 : int_spec x 0
| ZintPos n of x = n.+1 : int_spec x n.+1
| ZintNeg n of x = - (n.+1)%:Z : int_spec x (- n.+1).

Lemma intP x : int_spec x x. Proof. by move: x=> [] []; constructor. Qed.

Lemma addzC : commutative addz.
Proof. by move=> [] m [] n //=; rewrite addnC. Qed.

Lemma add0z : left_id 0 addz. Proof. by move=> [] [|]. Qed.

Lemma oppzK : involutive oppz. Proof. by do 2?case. Qed.

Lemma oppz_add : {morph oppz : m n / m + n}.
Proof.
move=> [[|n]|n] [[|m]|m] /=; rewrite ?NegzE ?oppzK ?addnS ?addn0 ?subn0 //;
  rewrite ?ltnS[m <= n]leqNgt [n <= m]leqNgt; case: ltngtP=> hmn /=;
    by rewrite ?hmn ?subnn // ?oppzK ?subSS ?subnS ?prednK // ?subn_gt0.
Qed.

Lemma add1Pz (n : int) : 1 + (n - 1) = n.
Proof. by case: (intP n)=> // n' /= _; rewrite ?(subn1, addn0). Qed.

Lemma subSz1 (n : int) : 1 + n - 1 = n.
Proof.
by apply: (inv_inj oppzK); rewrite addzC !oppz_add oppzK [_ - n]addzC add1Pz.
Qed.

Lemma addSnz (m : nat) (n : int) : (m.+1%N) + n = 1 + (m + n).
Proof.
move: m n=> [|m] [] [|n] //=; rewrite ?add1n ?subn1 // !(ltnS, subSS).
rewrite [n <= m]leqNgt; case: ltngtP=> hmn /=; rewrite ?hmn ?subnn //.
  by rewrite subnS add1n prednK ?subn_gt0.
by rewrite ltnS leqn0 subn_eq0 leqNgt hmn /= subnS subn1.
Qed.

Lemma addSz (m n : int) : (1 + m) + n = 1 + (m + n).
Proof.
case: m => [] m; first by rewrite -PoszD add1n addSnz.
rewrite !NegzE; apply: (inv_inj oppzK).
rewrite !oppz_add !oppzK addSnz [-1%:Z + _]addzC addSnz add1Pz.
by rewrite [-1%:Z + _]addzC subSz1.
Qed.

Lemma addPz (m n : int) : (m - 1) + n = (m + n) - 1.
Proof.
by apply: (inv_inj oppzK); rewrite !oppz_add oppzK [_ + 1]addzC addSz addzC.
Qed.

Lemma addzA : associative addz.
Proof.
elim=> [|m ihm|m ihm] n p; first by rewrite !add0z.
  by rewrite -add1n PoszD !addSz ihm.
by rewrite -add1n addnC PoszD oppz_add !addPz ihm.
Qed.

Lemma addNz : left_inverse (0:int) oppz addz. Proof. by do 3?elim. Qed.

Lemma predn_int (n : nat) : 0 < n -> n.-1%:Z = n - 1.
Proof. by case: n=> // n _ /=; rewrite subn1. Qed.

Definition Mixin := ZmodMixin addzA addzC add0z addNz.

End intZmod.
End intZmod.

Canonical int_ZmodType := ZmodType int intZmod.Mixin.

Local Open Scope ring_scope.

Section intZmoduleTheory.

Local Coercion Posz : nat >-> int.

Lemma PoszD : {morph Posz : n m / (n + m)%N >-> n + m}. Proof. by []. Qed.

Lemma NegzE (n : nat) : Negz n = -(n.+1)%:Z. Proof. by []. Qed.

Lemma int_rect (P : int -> Type) :
  P 0 -> (forall n : nat, P n -> P (n.+1)%N)
  -> (forall n : nat, P (- (n%:Z)) -> P (- (n.+1%N%:Z)))
  -> forall n : int, P n.
Proof.
by move=> P0 hPp hPn []; elim=> [|n ihn]//; do ?[apply: hPn | apply: hPp].
Qed.

Definition int_rec := int_rect.
Definition int_ind := int_rect.

CoInductive int_spec (x : int) : int -> Type :=
| ZintNull : int_spec x 0
| ZintPos n : int_spec x n.+1
| ZintNeg n : int_spec x (- (n.+1)%:Z).

Lemma intP x : int_spec x x.
Proof. by move: x=> [] [] *; rewrite ?NegzE; constructor. Qed.

Definition oppz_add := (@opprD [zmodType of int]).

Lemma subzn (m n : nat) : (n <= m)%N -> m%:Z - n%:Z = (m - n)%N.
Proof.
elim: n=> //= [|n ihn] hmn; first by rewrite subr0 subn0.
rewrite subnS -addn1 !PoszD opprD addrA ihn 1?ltnW //.
by rewrite intZmod.predn_int // subn_gt0.
Qed.

Lemma subzSS (m n : nat) : m.+1%:Z - n.+1%:Z = m%:Z - n%:Z.
Proof. by elim: n m=> [|n ihn] m //; rewrite !subzn. Qed.

End intZmoduleTheory.

Module intRing.
Section intRing.

Local Coercion Posz : nat >-> int.

Definition mulz (m n : int) :=
  match m, n with
    | Posz m', Posz n' => (m' * n')%N%:Z
    | Negz m', Negz n' => (m'.+1%N * n'.+1%N)%N%:Z
    | Posz m', Negz n' => - (m' * (n'.+1%N))%N%:Z
    | Negz n', Posz m' => - (m' * (n'.+1%N))%N%:Z
  end.

Local Notation "1" := (1%N:int) : int_scope.
Local Notation "*%Z" := (@mulz) : int_scope.
Local Notation "x * y" := (mulz x y) : int_scope.

Lemma mul0z : left_zero 0 *%Z.
Proof. by case=> [n|[|n]] //=; rewrite muln0. Qed.

Lemma mulzC : commutative mulz.
Proof. by move=> [] m [] n //=; rewrite mulnC. Qed.

Lemma mulz0 : right_zero 0 *%Z.
Proof. by move=> x; rewrite mulzC mul0z. Qed.

Lemma mulzN (m n : int) : (m * (- n))%Z = - (m * n)%Z.
Proof.
by case: (intP m)=> {m} [|m|m]; rewrite ?mul0z //;
case: (intP n)=> {n} [|n|n]; rewrite ?mulz0 //= mulnC.
Qed.

Lemma mulNz (m n : int) : ((- m) * n)%Z = - (m * n)%Z.
Proof. by rewrite mulzC mulzN mulzC. Qed.

Lemma mulzA : associative mulz.
Proof.
by move=> [] m [] n [] p; rewrite ?NegzE ?(mulnA,mulNz,mulzN,opprK) //= ?mulnA.
Qed.

Lemma mul1z : left_id 1%Z mulz.
Proof. by case=> [[|n]|n] //=; rewrite ?mul1n// plusE addn0. Qed.

Lemma mulzS (x : int) (n : nat) : (x * n.+1%:Z)%Z = x + (x * n)%Z.
Proof.
by case: (intP x)=> [|m'|m'] //=; [rewrite mulnS|rewrite mulSn -opprD].
Qed.

Lemma mulz_addl : left_distributive mulz (+%R).
Proof.
move=> x y z; elim: z=> [|n|n]; first by rewrite !(mul0z,mulzC).
  by rewrite !mulzS=> ->; rewrite !addrA [X in X + _]addrAC.
rewrite !mulzN !mulzS -!opprD=> /(inv_inj (@opprK _))->.
by rewrite !addrA [X in X + _]addrAC.
Qed.

Lemma nonzero1z : 1%Z != 0. Proof. by []. Qed.

Definition comMixin := ComRingMixin mulzA mulzC mul1z mulz_addl nonzero1z.

End intRing.
End intRing.

Canonical int_Ring := Eval hnf in RingType int intRing.comMixin.
Canonical int_comRing := Eval hnf in ComRingType int intRing.mulzC.

Section intRingTheory.

Implicit Types m n : int.
Local Coercion Posz : nat >-> int.

Lemma PoszM : {morph Posz : n m / (n * m)%N >-> n * m}. Proof. by []. Qed.

Lemma intS (n : nat) : n.+1%:Z = 1 + n%:Z. Proof. by rewrite -PoszD. Qed.

Lemma predn_int (n : nat) : (0 < n)%N -> n.-1%:Z = n%:Z - 1.
Proof. exact: intZmod.predn_int. Qed.

End intRingTheory.

Module intUnitRing.
Section intUnitRing.
Implicit Types m n : int.
Local Coercion Posz : nat >-> int.

Definition unitz := [qualify a n : int | (n == 1) || (n == -1)].
Definition invz n : int := n.

Lemma mulVz : {in unitz, left_inverse 1%R invz *%R}.
Proof. by move=> n /pred2P[] ->. Qed.

Lemma mulzn_eq1 m (n : nat) : (m * n == 1) = (m == 1) && (n == 1%N).
Proof. by case: m=> m /=; [rewrite -PoszM [_==_]muln_eq1 | case: n]. Qed.

Lemma unitzPl m n : n * m = 1 -> m \is a unitz.
Proof.
case: m => m; move/eqP; rewrite qualifE.
* by rewrite mulzn_eq1; case/andP=> _; move/eqP->.
* by rewrite NegzE intS mulrN -mulNr mulzn_eq1; case/andP=> _.
Qed.

Lemma  invz_out : {in [predC unitz], invz =1 id}.
Proof. exact. Qed.

Lemma idomain_axiomz m n : m * n = 0 -> (m == 0) || (n == 0).
Proof.
by case: m n => m [] n //=; move/eqP; rewrite ?(NegzE,mulrN,mulNr);
  rewrite ?(inv_eq (@opprK _)) -PoszM [_==_]muln_eq0.
Qed.

Definition comMixin := ComUnitRingMixin mulVz unitzPl invz_out.

End intUnitRing.
End intUnitRing.

Canonical int_unitRingType :=
  Eval hnf in UnitRingType int intUnitRing.comMixin.
Canonical int_comUnitRing := Eval hnf in [comUnitRingType of int].
Canonical int_iDomain :=
  Eval hnf in IdomainType int intUnitRing.idomain_axiomz.

Definition absz m := match m with Posz p => p | Negz n => n.+1 end.
Notation "m - n" :=
  (@GRing.add int_ZmodType m%N (@GRing.opp int_ZmodType n%N)) : distn_scope.
Arguments Scope absz [distn_scope].
Local Notation "`| m |" := (absz m) : nat_scope.

Module intOrdered.
Section intOrdered.
Implicit Types m n p : int.
Local Coercion Posz : nat >-> int.

Local Notation normz m := (absz m)%:Z.

Definition lez m n :=
  match m, n with
    | Posz m', Posz n' => (m' <= n')%N
    | Posz m', Negz n' => false
    | Negz m', Posz n' => true
    | Negz m', Negz n' => (n' <= m')%N
  end.

Definition ltz m n :=
  match m, n with
    | Posz m', Posz n' => (m' < n')%N
    | Posz m', Negz n' => false
    | Negz m', Posz n' => true
    | Negz m', Negz n' => (n' < m')%N
  end.

Fact lez_norm_add x y : lez (normz (x + y)) (normz x + normz y).
Proof.
move: x y=> [] m [] n; rewrite /= ?addnS //=;
rewrite /GRing.add /GRing.Zmodule.add /=; case: ltnP=> //=;
rewrite ?addSn ?ltnS ?leq_subLR ?(addnS, addSn) ?(leq_trans _ (leqnSn _)) //;
by rewrite 1?addnCA ?leq_addr ?addnA ?leq_addl.
Qed.

Fact ltz_add x y : ltz 0 x -> ltz 0 y -> ltz 0 (x + y).
Proof. by move: x y => [] x [] y //= hx hy; rewrite ltn_addr. Qed.

Fact eq0_normz x : normz x = 0 -> x = 0. Proof. by case: x. Qed.

Fact lez_total x y : lez x y || lez y x.
Proof. by move: x y => [] x [] y //=; apply: leq_total. Qed.

Lemma abszN (n : nat) : absz (- n%:Z) = n. Proof. by case: n. Qed.

Fact normzM : {morph (fun n => normz n) : x y / x * y}.
Proof. by move=> [] x [] y; rewrite // abszN // mulnC. Qed.

Lemma subz_ge0 m n : lez 0 (n - m) = lez m n.
Proof.
case: (intP m); case: (intP n)=> // {m n} m n /=;
rewrite ?ltnS -?opprD ?opprB ?subzSS; case: leqP=> // hmn;
by [ rewrite subzn //
   | rewrite -opprB subzn ?(ltnW hmn) //;
      move: hmn; rewrite -subn_gt0; case: (_ - _)%N].
Qed.

Fact lez_def x y : (lez x y) = (normz (y - x) == y - x).
Proof. by rewrite -subz_ge0; move: (_ - _) => [] n //=; rewrite eqxx. Qed.

Fact ltz_def x y : (ltz x y) = (y != x) && (lez x y).
Proof.
by move: x y=> [] x [] y //=; rewrite (ltn_neqAle, leq_eqVlt) // eq_sym.
Qed.

Definition Mixin :=
   NumMixin lez_norm_add ltz_add eq0_normz (in2W lez_total) normzM
            lez_def ltz_def.

End intOrdered.
End intOrdered.

Canonical int_numDomainType := NumDomainType int intOrdered.Mixin.
Canonical int_realDomainType := RealDomainType int (intOrdered.lez_total 0).

Section intOrderedTheory.

Local Coercion Posz : nat >-> int.
Implicit Types m n p : nat.
Implicit Types x y z : int.

Lemma lez_nat m n : (m <= n :> int) = (m <= n)%N.
Proof. by []. Qed.

Lemma ltz_nat  m n : (m < n :> int) = (m < n)%N.
Proof. by rewrite ltnNge ltrNge lez_nat. Qed.

Definition ltez_nat := (lez_nat, ltz_nat).

Lemma leNz_nat m n : (- m%:Z <= n). Proof. by case: m. Qed.

Lemma ltNz_nat m n : (- m%:Z < n) = (m != 0%N) || (n != 0%N).
Proof. by move: m n=> [|?] []. Qed.

Definition lteNz_nat := (leNz_nat, ltNz_nat).

Lemma lezN_nat m n : (m%:Z <= - n%:Z) = (m == 0%N) && (n == 0%N).
Proof. by move: m n=> [|?] []. Qed.

Lemma ltzN_nat m n : (m%:Z < - n%:Z) = false.
Proof. by move: m n=> [|?] []. Qed.

Lemma le0z_nat n : 0 <= n :> int. Proof. by []. Qed.

Lemma lez0_nat n : n <= 0 :> int = (n == 0%N :> nat). Proof. by elim: n. Qed.

Definition ltezN_nat := (lezN_nat, ltzN_nat).
Definition ltez_natE := (ltez_nat, lteNz_nat, ltezN_nat, le0z_nat, lez0_nat).

Lemma gtz0_ge1 x : (0 < x) = (1 <= x). Proof. by case: (intP x). Qed.

Lemma lez_add1r x y : (1 + x <= y) = (x < y).
Proof. by rewrite -subr_gt0 gtz0_ge1 lter_sub_addr. Qed.

Lemma lez_addr1 x y : (x + 1 <= y) = (x < y).
Proof. by rewrite addrC lez_add1r. Qed.

Lemma ltz_add1r x y : (x < 1 + y) = (x <= y).
Proof. by rewrite -lez_add1r ler_add2l. Qed.

Lemma ltz_addr1 x y : (x < y + 1) = (x <= y).
Proof. by rewrite -lez_addr1 ler_add2r. Qed.

End intOrderedTheory.

Bind Scope ring_scope with int.

(* definition of intmul *)
Definition intmul (R : zmodType) (x : R) (n : int) := nosimpl
  match n with
    | Posz n => (x *+ n)%R
    | Negz n => (x *- (n.+1))%R
  end.

Notation "*~%R" := (@intmul _) (at level 0, format " *~%R") : ring_scope.
Notation "x *~ n" := (intmul x n)
  (at level 40, left associativity, format "x  *~  n") : ring_scope.
Notation intr := ( *~%R 1).
Notation "n %:~R" := (1 *~ n)%R
  (at level 2, left associativity, format "n %:~R")  : ring_scope.

Lemma pmulrn (R : zmodType) (x : R) (n : nat) : x *+ n = x *~ n%:Z.
Proof. by []. Qed.

Lemma nmulrn (R : zmodType) (x : R) (n : nat) : x *- n = x *~ - n%:Z.
Proof. by case: n=> [] //; rewrite ?oppr0. Qed.

Section ZintLmod.

Definition zmodule (M : Type) : Type := M.
Local Notation "M ^z" := (zmodule M) (at level 2, format "M ^z") : type_scope.
Local Coercion Posz : nat >-> int.

Variable M : zmodType.

Implicit Types m n : int.
Implicit Types x y z : M.

Fact mulrzA_C m n x : (x *~ n) *~ m = x *~ (m * n).
Proof.
elim: m=> [|m _|m _]; elim: n=> [|n _|n _]; rewrite /intmul //=;
rewrite ?(muln0, mulr0n, mul0rn, oppr0, mulNrn, opprK) //;
 do ?by rewrite mulnC mulrnA.
* by rewrite -mulrnA mulnC.
* by rewrite -mulrnA.
Qed.

Fact mulrzAC m n x : (x *~ n) *~ m = (x *~ m) *~ n.
Proof. by rewrite !mulrzA_C mulrC. Qed.

Fact mulr1z (x : M) : x *~ 1 = x. Proof. by []. Qed.

Fact mulrzDr m : {morph ( *~%R^~ m : M -> M) : x y / x + y}.
Proof.
by elim: m=> [|m _|m _] x y;
  rewrite ?addr0 /intmul //= ?mulrnDl // opprD.
Qed.

Lemma mulrzBl_nat (m n : nat) x : x *~ (m%:Z - n%:Z) = x *~ m - x *~ n.
Proof.
case: (leqP m n)=> hmn; rewrite /intmul //=.
  rewrite addrC -{1}[m:int]opprK -opprD subzn //.
  rewrite -{2}[n](@subnKC m)// mulrnDr opprD addrA subrr sub0r.
  by case hdmn: (_ - _)%N=> [|dmn] /=; first by rewrite mulr0n oppr0.
have hnm := ltnW hmn.
rewrite -{2}[m](@subnKC n)// mulrnDr addrAC subrr add0r.
by rewrite subzn.
Qed.

Fact mulrzDl x : {morph *~%R x : m n / m + n}.
Proof.
elim=> [|m _|m _]; elim=> [|n _|n _]; rewrite /intmul //=;
rewrite -?(opprD) ?(add0r, addr0, mulrnDr, subn0) //.
* by rewrite -/(intmul _ _) mulrzBl_nat.
* by rewrite -/(intmul _ _) addrC mulrzBl_nat addrC.
* by rewrite -addnS -addSn mulrnDr.
Qed.

Definition Mint_LmodMixin :=
  @LmodMixin _ [zmodType of M] (fun n x => x *~ n)
   mulrzA_C mulr1z mulrzDr mulrzDl.
Canonical Mint_LmodType := LmodType int M^z Mint_LmodMixin.

Lemma scalezrE n x : n *: (x : M^z) = x *~ n. Proof. by []. Qed.

Lemma mulrzA x m n :  x *~ (m * n) = x *~ m *~ n.
Proof. by rewrite -!scalezrE scalerA mulrC. Qed.

Lemma mulr0z x : x *~ 0 = 0. Proof. by []. Qed.

Lemma mul0rz n : 0 *~ n = 0 :> M.
Proof. by rewrite -scalezrE scaler0. Qed.

Lemma mulrNz x n : x *~ (- n) = - (x *~ n).
Proof. by rewrite -scalezrE scaleNr. Qed.

Lemma mulrN1z x : x *~ (- 1) = - x. Proof. by rewrite -scalezrE scaleN1r. Qed.

Lemma mulNrz x n : (- x) *~ n = - (x *~ n).
Proof. by rewrite -scalezrE scalerN. Qed.

Lemma mulrzBr x m n : x *~ (m - n) = x *~ m - x *~ n.
Proof. by rewrite -scalezrE scalerBl. Qed.

Lemma mulrzBl x y n : (x - y) *~ n = x *~ n - y *~ n.
Proof. by rewrite -scalezrE scalerBr. Qed.

Lemma mulrz_nat (n : nat) x : x *~ n%:R = x *+ n.
Proof. by rewrite -scalezrE scaler_nat. Qed.

Lemma mulrz_sumr : forall x I r (P : pred I) F,
  x *~ (\sum_(i <- r | P i) F i) = \sum_(i <- r | P i) x *~ F i.
Proof. by rewrite -/M^z; apply: scaler_suml. Qed.

Lemma mulrz_suml : forall n I r (P : pred I) (F : I -> M),
  (\sum_(i <- r | P i) F i) *~ n= \sum_(i <- r | P i) F i *~ n.
Proof. by rewrite -/M^z; apply: scaler_sumr. Qed.

Canonical intmul_additive x := Additive (@mulrzBr x).

End ZintLmod.

Lemma ffunMzE (I : finType) (M : zmodType) (f : {ffun I -> M}) z x :
  (f *~ z) x = f x *~ z.
Proof. by case: z => n; rewrite ?ffunE ffunMnE. Qed.

Lemma intz (n : int) : n%:~R = n.
Proof.
elim: n=> //= n ihn; rewrite /intmul /=.
  by rewrite -addn1 mulrnDr /= PoszD -ihn.
by rewrite nmulrn intS opprD mulrzDl ihn.
Qed.

Lemma natz (n : nat) : n%:R = n%:Z :> int.
Proof. by rewrite pmulrn intz. Qed.

Section RintMod.

Local Coercion Posz : nat >-> int.
Variable R : ringType.

Implicit Types m n : int.
Implicit Types x y z : R.

Lemma mulrzAl n x y : (x *~ n) * y = (x * y) *~ n.
Proof.
by elim: n=> //= *; rewrite ?mul0r ?mulr0z // /intmul /= -mulrnAl -?mulNr.
Qed.

Lemma mulrzAr n x y : x * (y *~ n) = (x * y) *~ n.
Proof.
by elim: n=> //= *; rewrite ?mulr0 ?mulr0z // /intmul /= -mulrnAr -?mulrN.
Qed.

Lemma mulrzl x n : n%:~R * x = x *~ n.
Proof. by rewrite mulrzAl mul1r. Qed.

Lemma mulrzr x n : x * n%:~R = x *~ n.
Proof. by rewrite mulrzAr mulr1. Qed.

Lemma mulNrNz n x : (-x) *~ (-n) = x *~ n.
Proof. by rewrite mulNrz mulrNz opprK. Qed.

Lemma mulrbz x (b : bool) : x *~ b = (if b then x else 0).
Proof. by case: b. Qed.

Lemma intrD m n : (m + n)%:~R = m%:~R + n%:~R :> R.
Proof. exact: mulrzDl. Qed.

Lemma intrM m n : (m * n)%:~R = m%:~R * n%:~R :> R.
Proof. by rewrite mulrzA -mulrzr. Qed.

Lemma intmul1_is_rmorphism : rmorphism ( *~%R (1 : R)).
Proof.
by do ?split; move=> // x y /=; rewrite ?intrD ?mulrNz ?intrM.
Qed.
Canonical intmul1_rmorphism := RMorphism intmul1_is_rmorphism.

Lemma mulr2z n : n *~ 2 = n + n. Proof. exact: mulr2n. Qed.

End RintMod.

Lemma mulrzz m n : m *~ n = m * n. Proof. by rewrite -mulrzr intz. Qed.

Lemma mulz2 n : n * 2%:Z = n + n. Proof. by rewrite -mulrzz. Qed.

Lemma mul2z n : 2%:Z * n = n + n. Proof. by rewrite mulrC -mulrzz. Qed.

Section LMod.

Variable R : ringType.
Variable V : (lmodType R).
Local Coercion Posz : nat >-> int.

Implicit Types m n : int.
Implicit Types x y z : R.
Implicit Types u v w : V.

Lemma scaler_int n v : n%:~R *: v = v *~ n.
Proof.
elim: n=> [|n ihn|n ihn]; first by rewrite scale0r.
  by rewrite intS !mulrzDl scalerDl ihn scale1r.
by rewrite intS opprD !mulrzDl scalerDl ihn scaleN1r.
Qed.

Lemma scalerMzl a v n : (a *: v) *~ n = (a *~ n) *: v.
Proof. by rewrite -mulrzl -scaler_int scalerA. Qed.

Lemma scalerMzr a v n : (a *: v) *~ n = a *: (v *~ n).
Proof. by rewrite -!scaler_int !scalerA mulrzr mulrzl. Qed.

End LMod.

Lemma mulrz_int (M : zmodType) (n : int) (x : M) : x *~ n%:~R = x *~ n.
Proof. by rewrite -scalezrE scaler_int. Qed.

Section MorphTheory.
Local Coercion Posz : nat >-> int.
Section Additive.
Variables (U V : zmodType) (f : {additive U -> V}).

Lemma raddfMz n : {morph f : x / x *~ n}.
Proof.
case: n=> n x /=; first exact: raddfMn.
by rewrite NegzE !mulrNz; apply: raddfMNn.
Qed.

End Additive.

Section Multiplicative.

Variables (R S : ringType) (f : {rmorphism R -> S}).

Lemma rmorphMz : forall n, {morph f : x / x *~ n}. Proof. exact: raddfMz. Qed.

Lemma rmorph_int : forall n, f n%:~R = n%:~R.
Proof. by move=> n; rewrite rmorphMz rmorph1. Qed.

End Multiplicative.

Section Linear.

Variable R : ringType.
Variables (U V : lmodType R) (f : {linear U -> V}).

Lemma linearMn : forall n, {morph f : x / x *~ n}. Proof. exact: raddfMz. Qed.

End Linear.

Lemma raddf_int_scalable (aV rV : lmodType int) (f : {additive aV -> rV}) :
  scalable f.
Proof. by move=> z u; rewrite -[z]intz !scaler_int raddfMz. Qed.

Section Zintmul1rMorph.

Variable R : ringType.

Lemma commrMz (x y : R) n : GRing.comm x y -> GRing.comm x (y *~ n).
Proof. by rewrite /GRing.comm=> com_xy; rewrite mulrzAr mulrzAl com_xy. Qed.

Lemma commr_int (x : R) n : GRing.comm x n%:~R.
Proof. by apply: commrMz; apply: commr1. Qed.

End Zintmul1rMorph.

Section ZintBigMorphism.

Variable R : ringType.

Lemma sumMz : forall I r (P : pred I) F,
 (\sum_(i <- r | P i) F i)%N%:~R = \sum_(i <- r | P i) ((F i)%:~R) :> R.
Proof. by apply: big_morph=> // x y; rewrite !pmulrn -rmorphD. Qed.

Lemma prodMz : forall I r (P : pred I) F,
 (\prod_(i <- r | P i) F i)%N%:~R = \prod_(i <- r | P i) ((F i)%:~R) :> R.
Proof. by apply: big_morph=> // x y; rewrite !pmulrn PoszM -rmorphM. Qed.

End ZintBigMorphism.

Section Frobenius.

Variable R : ringType.
Implicit Types x y : R.

Variable p : nat.
Hypothesis charFp : p \in [char R].

Local Notation "x ^f" := (Frobenius_aut charFp x).

Lemma Frobenius_autMz x n : (x *~ n)^f = x^f *~ n.
Proof.
case: n=> n /=; first exact: Frobenius_autMn.
by rewrite !NegzE !mulrNz Frobenius_autN Frobenius_autMn.
Qed.

Lemma Frobenius_aut_int n : (n%:~R)^f = n%:~R.
Proof. by rewrite Frobenius_autMz Frobenius_aut1. Qed.

End Frobenius.

Section NumMorphism.

Section PO.

Variables (R : numDomainType).

Implicit Types n m : int.
Implicit Types x y : R.

Lemma rmorphzP (f : {rmorphism int -> R}) : f =1 ( *~%R 1).
Proof.
move=> n; wlog : n / 0 <= n; case: n=> [] n //; do ?exact.
  by rewrite NegzE !rmorphN=>->.
move=> _; elim: n=> [|n ihn]; first by rewrite rmorph0.
by rewrite intS !rmorphD !rmorph1 ihn.
Qed.

(* intmul and ler/ltr *)
Lemma ler_pmulz2r n (hn : 0 < n) : {mono *~%R^~ n :x y / x <= y :> R}.
Proof. by move=> x y; case: n hn=> [[]|] // n _; rewrite ler_pmuln2r. Qed.

Lemma ltr_pmulz2r n (hn : 0 < n) : {mono *~%R^~ n : x y / x < y :> R}.
Proof. exact: lerW_mono (ler_pmulz2r _). Qed.

Lemma ler_nmulz2r n (hn : n < 0) : {mono *~%R^~ n : x y /~ x <= y :> R}.
Proof.
move=> x y /=; rewrite -![_ *~ n]mulNrNz.
by rewrite ler_pmulz2r (oppr_cp0, ler_opp2).
Qed.

Lemma ltr_nmulz2r n (hn : n < 0) : {mono *~%R^~ n : x y /~ x < y :> R}.
Proof. exact: lerW_nmono (ler_nmulz2r _). Qed.

Lemma ler_wpmulz2r n (hn : 0 <= n) : {homo *~%R^~ n : x y / x <= y :> R}.
Proof. by move=> x y xy; case: n hn=> [] // n _; rewrite ler_wmuln2r. Qed.

Lemma ler_wnmulz2r n (hn : n <= 0) : {homo *~%R^~ n : x y /~ x <= y :> R}.
Proof.
by move=> x y xy /=; rewrite -ler_opp2 -!mulrNz ler_wpmulz2r // oppr_ge0.
Qed.

Lemma mulrz_ge0 x n (x0 : 0 <= x) (n0 : 0 <= n) : 0 <= x *~ n.
Proof. by rewrite -(mul0rz _ n) ler_wpmulz2r. Qed.

Lemma mulrz_le0 x n (x0 : x <= 0) (n0 : n <= 0) : 0 <= x *~ n.
Proof. by rewrite -(mul0rz _ n) ler_wnmulz2r. Qed.

Lemma mulrz_ge0_le0 x n (x0 : 0 <= x) (n0 : n <= 0) : x *~ n <= 0.
Proof. by rewrite -(mul0rz _ n) ler_wnmulz2r. Qed.

Lemma mulrz_le0_ge0 x n (x0 : x <= 0) (n0 : 0 <= n) : x *~ n <= 0.
Proof. by rewrite -(mul0rz _ n) ler_wpmulz2r. Qed.

Lemma pmulrz_lgt0 x n (n0 : 0 < n) : 0 < x *~ n = (0 < x).
Proof. by rewrite -(mul0rz _ n) ltr_pmulz2r // mul0rz. Qed.

Lemma nmulrz_lgt0 x n (n0 : n < 0) : 0 < x *~ n = (x < 0).
Proof. by rewrite -(mul0rz _ n) ltr_nmulz2r // mul0rz. Qed.

Lemma pmulrz_llt0 x n (n0 : 0 < n) : x *~ n < 0 = (x < 0).
Proof. by rewrite -(mul0rz _ n) ltr_pmulz2r // mul0rz. Qed.

Lemma nmulrz_llt0 x n (n0 : n < 0) : x *~ n < 0 = (0 < x).
Proof. by rewrite -(mul0rz _ n) ltr_nmulz2r // mul0rz. Qed.

Lemma pmulrz_lge0 x n (n0 : 0 < n) : 0 <= x *~ n = (0 <= x).
Proof. by rewrite -(mul0rz _ n) ler_pmulz2r // mul0rz. Qed.

Lemma nmulrz_lge0 x n (n0 : n < 0) : 0 <= x *~ n = (x <= 0).
Proof. by rewrite -(mul0rz _ n) ler_nmulz2r // mul0rz. Qed.

Lemma pmulrz_lle0 x n (n0 : 0 < n) : x *~ n <= 0 = (x <= 0).
Proof. by rewrite -(mul0rz _ n) ler_pmulz2r // mul0rz. Qed.

Lemma nmulrz_lle0 x n (n0 : n < 0) : x *~ n <= 0 = (0 <= x).
Proof. by rewrite -(mul0rz _ n) ler_nmulz2r // mul0rz. Qed.

Lemma ler_wpmulz2l x (hx : 0 <= x) : {homo *~%R x : x y / x <= y}.
Proof.
by move=> m n /= hmn; rewrite -subr_ge0 -mulrzBr mulrz_ge0 // subr_ge0.
Qed.

Lemma ler_wnmulz2l x (hx : x <= 0) : {homo *~%R x : x y /~ x <= y}.
Proof.
by move=> m n /= hmn; rewrite -subr_ge0 -mulrzBr mulrz_le0 // subr_le0.
Qed.

Lemma ler_pmulz2l x (hx : 0 < x) : {mono *~%R x : x y / x <= y}.
Proof.
move=> m n /=; rewrite real_mono ?num_real // => {m n}.
by move=> m n /= hmn; rewrite -subr_gt0 -mulrzBr pmulrz_lgt0 // subr_gt0.
Qed.

Lemma ler_nmulz2l x (hx : x < 0) : {mono *~%R x : x y /~ x <= y}.
Proof.
move=> m n /=; rewrite real_nmono ?num_real // => {m n}.
by move=> m n /= hmn; rewrite -subr_gt0 -mulrzBr nmulrz_lgt0 // subr_lt0.
Qed.

Lemma ltr_pmulz2l x (hx : 0 < x) : {mono *~%R x : x y / x < y}.
Proof. exact: lerW_mono (ler_pmulz2l _). Qed.

Lemma ltr_nmulz2l x (hx : x < 0) : {mono *~%R x : x y /~ x < y}.
Proof. exact: lerW_nmono (ler_nmulz2l _). Qed.

Lemma pmulrz_rgt0 x n (x0 : 0 < x) : 0 < x *~ n = (0 < n).
Proof. by rewrite -(mulr0z x) ltr_pmulz2l. Qed.

Lemma nmulrz_rgt0 x n (x0 : x < 0) : 0 < x *~ n = (n < 0).
Proof. by rewrite -(mulr0z x) ltr_nmulz2l. Qed.

Lemma pmulrz_rlt0 x n (x0 : 0 < x) : x *~ n < 0 = (n < 0).
Proof. by rewrite -(mulr0z x) ltr_pmulz2l. Qed.

Lemma nmulrz_rlt0 x n (x0 : x < 0) : x *~ n < 0 = (0 < n).
Proof. by rewrite -(mulr0z x) ltr_nmulz2l. Qed.

Lemma pmulrz_rge0 x n (x0 : 0 < x) : 0 <= x *~ n = (0 <= n).
Proof. by rewrite -(mulr0z x) ler_pmulz2l. Qed.

Lemma nmulrz_rge0 x n (x0 : x < 0) : 0 <= x *~ n = (n <= 0).
Proof. by rewrite -(mulr0z x) ler_nmulz2l. Qed.

Lemma pmulrz_rle0 x n (x0 : 0 < x) : x *~ n <= 0 = (n <= 0).
Proof. by rewrite -(mulr0z x) ler_pmulz2l. Qed.

Lemma nmulrz_rle0 x n (x0 : x < 0) : x *~ n <= 0 = (0 <= n).
Proof. by rewrite -(mulr0z x) ler_nmulz2l. Qed.

Lemma mulrIz x (hx : x != 0) : injective ( *~%R x).
Proof.
move=> y z; rewrite -![x *~ _]mulrzr => /(mulfI hx).
by apply: mono_inj y z; apply: ler_pmulz2l.
Qed.

Lemma ler_int m n : (m%:~R <= n%:~R :> R) = (m <= n).
Proof. by rewrite ler_pmulz2l. Qed.

Lemma ltr_int m n : (m%:~R < n%:~R :> R) = (m < n).
Proof. by rewrite ltr_pmulz2l. Qed.

Lemma eqr_int m n : (m%:~R == n%:~R :> R) = (m == n).
Proof. by rewrite (inj_eq (mulrIz _)) ?oner_eq0. Qed.

Lemma ler0z n : (0 <= n%:~R :> R) = (0 <= n).
Proof. by rewrite pmulrz_rge0. Qed.

Lemma ltr0z n : (0 < n%:~R :> R) = (0 < n).
Proof. by rewrite pmulrz_rgt0. Qed.

Lemma lerz0 n : (n%:~R <= 0 :> R) = (n <= 0).
Proof. by rewrite pmulrz_rle0. Qed.

Lemma ltrz0 n : (n%:~R < 0 :> R) = (n < 0).
Proof. by rewrite pmulrz_rlt0. Qed.

Lemma ler1z (n : int) : (1 <= n%:~R :> R) = (1 <= n).
Proof. by rewrite -[1]/(1%:~R) ler_int. Qed.

Lemma ltr1z (n : int) : (1 < n%:~R :> R) = (1 < n).
Proof. by rewrite -[1]/(1%:~R) ltr_int. Qed.

Lemma lerz1 n : (n%:~R <= 1 :> R) = (n <= 1).
Proof. by rewrite -[1]/(1%:~R) ler_int. Qed.

Lemma ltrz1 n : (n%:~R < 1 :> R) = (n < 1).
Proof. by rewrite -[1]/(1%:~R) ltr_int. Qed.

Lemma intr_eq0 n : (n%:~R == 0 :> R) = (n == 0).
Proof. by rewrite -(mulr0z 1) (inj_eq (mulrIz _)) // oner_eq0. Qed.

Lemma mulrz_eq0 x n : (x *~ n == 0) = ((n == 0) || (x == 0)).
Proof. by rewrite -mulrzl mulf_eq0 intr_eq0. Qed.

Lemma mulrz_neq0 x n : x *~ n != 0 = ((n != 0) && (x != 0)).
Proof. by rewrite mulrz_eq0 negb_or. Qed.

Lemma realz n : (n%:~R : R) \in Num.real.
Proof. by rewrite -topredE /Num.real /= ler0z lerz0 ler_total. Qed.
Hint Resolve realz.

Definition intr_inj := @mulrIz 1 (oner_neq0 R).

End PO.

End NumMorphism.

End MorphTheory.

Arguments intr_inj {R} [x1 x2].

Definition exprz (R : unitRingType) (x : R) (n : int) := nosimpl
  match n with
    | Posz n => x ^+ n
    | Negz n => x ^- (n.+1)
  end.

Notation "x ^ n" := (exprz x n) : ring_scope.

Section ExprzUnitRing.

Variable R : unitRingType.
Implicit Types x y : R.
Implicit Types m n : int.
Local Coercion Posz : nat >-> int.

Lemma exprnP x (n : nat) : x ^+ n = x ^ n. Proof. by []. Qed.

Lemma exprnN x (n : nat) : x ^- n = x ^ -n%:Z.
Proof. by case: n=> //; rewrite oppr0 expr0 invr1. Qed.

Lemma expr0z x : x ^ 0 = 1. Proof. by []. Qed.

Lemma expr1z x : x ^ 1 = x. Proof. by []. Qed.

Lemma exprN1 x : x ^ (-1) = x^-1. Proof. by []. Qed.

Lemma invr_expz x n : (x ^ n)^-1 = x ^ (- n).
Proof.
by case: (intP n)=> // [|m]; rewrite ?opprK ?expr0z ?invr1 // invrK.
Qed.

Lemma exprz_inv x n : (x^-1) ^ n = x ^ (- n).
Proof.
by case: (intP n)=> // m; rewrite -[_ ^ (- _)]exprVn ?opprK ?invrK.
Qed.

Lemma exp1rz n : 1 ^ n = 1 :> R.
Proof.
by case: (intP n)=> // m; rewrite -?exprz_inv ?invr1; apply: expr1n.
Qed.

Lemma exprSz x (n : nat) : x ^ n.+1 = x * x ^ n. Proof. exact: exprS. Qed.

Lemma exprSzr x (n : nat) : x ^ n.+1 = x ^ n * x.
Proof. exact: exprSr. Qed.

Fact exprzD_nat x (m n : nat) : x ^ (m%:Z + n) = x ^ m * x ^ n.
Proof. exact: exprD. Qed.

Fact exprzD_Nnat x (m n : nat) : x ^ (-m%:Z + -n%:Z) = x ^ (-m%:Z) * x ^ (-n%:Z).
Proof. by rewrite -opprD -!exprz_inv exprzD_nat. Qed.

Lemma exprzD_ss x m n : (0 <= m) && (0 <= n) || (m <= 0) && (n <= 0)
  ->  x ^ (m + n) = x ^ m * x ^ n.
Proof.
case: (intP m)=> {m} [|m|m]; case: (intP n)=> {n} [|n|n] //= _;
by rewrite ?expr0z ?mul1r ?exprzD_nat ?exprzD_Nnat ?sub0r ?addr0 ?mulr1.
Qed.

Lemma exp0rz n : 0 ^ n = (n == 0)%:~R :> R.
Proof. by case: (intP n)=> // m; rewrite -?exprz_inv ?invr0 exprSz mul0r. Qed.

Lemma commrXz x y n : GRing.comm x y -> GRing.comm x (y ^ n).
Proof.
rewrite /GRing.comm; elim: n x y=> [|n ihn|n ihn] x y com_xy //=.
* by rewrite expr0z mul1r mulr1.
* by rewrite -exprnP commrX //.
rewrite -exprz_inv -exprnP commrX //.
case: (boolP (y \is a GRing.unit))=> uy; last by rewrite invr_out.
by apply/eqP; rewrite (can2_eq (mulrVK _) (mulrK _)) // -mulrA com_xy mulKr.
Qed.

Lemma exprMz_comm x y n : x \is a GRing.unit -> y \is a GRing.unit ->
  GRing.comm x y -> (x * y) ^ n = x ^ n * y ^ n.
Proof.
move=> ux uy com_xy; elim: n => [|n _|n _]; first by rewrite expr0z mulr1.
  by rewrite -!exprnP exprMn_comm.
rewrite -!exprnN -!exprVn com_xy -exprMn_comm ?invrM//.
exact/commrV/commr_sym/commrV.
Qed.

Lemma commrXz_wmulls x y n :
  0 <= n -> GRing.comm x y -> (x * y) ^ n = x ^ n * y ^ n.
Proof.
move=> n0 com_xy; elim: n n0 => [|n _|n _] //; first by rewrite expr0z mulr1.
by rewrite -!exprnP exprMn_comm.
Qed.

Lemma unitrXz x n (ux : x \is a GRing.unit) : x ^ n \is a GRing.unit.
Proof.
case: (intP n)=> {n} [|n|n]; rewrite ?expr0z ?unitr1 ?unitrX //.
by rewrite -invr_expz unitrV unitrX.
Qed.

Lemma exprzDr x (ux : x \is a GRing.unit) m n : x ^ (m + n) = x ^ m * x ^ n.
Proof.
move: n m; apply: wlog_ler=> n m hnm.
  by rewrite addrC hnm commrXz //; apply: commr_sym; apply: commrXz.
case: (intP m) hnm=> {m} [|m|m]; rewrite ?mul1r ?add0r //;
 case: (intP n)=> {n} [|n|n _]; rewrite ?mulr1 ?addr0 //;
   do ?by rewrite exprzD_ss.
rewrite -invr_expz subzSS !exprSzr invrM ?unitrX // -mulrA mulVKr //.
case: (leqP n m)=> [|/ltnW] hmn; rewrite -{2}(subnK hmn) exprzD_nat -subzn //.
  by rewrite mulrK ?unitrX.
by rewrite invrM ?unitrXz // mulVKr ?unitrXz // -opprB -invr_expz.
Qed.

Lemma exprz_exp x m n : (x ^ m) ^ n = (x ^ (m * n)).
Proof.
wlog: n / 0 <= n.
  by case: n=> [n -> //|n]; rewrite ?NegzE mulrN -?invr_expz=> -> /=.
elim: n x m=> [|n ihn|n ihn] x m // _; first by rewrite mulr0 !expr0z.
rewrite exprSz ihn // intS mulrDr mulr1 exprzD_ss //.
by case: (intP m)=> // m'; rewrite ?oppr_le0 //.
Qed.

Lemma exprzAC x m n : (x ^ m) ^ n = (x ^ n) ^ m.
Proof. by rewrite !exprz_exp mulrC. Qed.

Lemma exprz_out x n (nux : x \isn't a GRing.unit) (hn : 0 <= n) :
  x ^ (- n) = x ^ n.
Proof. by case: (intP n) hn=> //= m; rewrite -exprnN -exprVn invr_out. Qed.

End ExprzUnitRing.

Section Exprz_Zint_UnitRing.

Variable R : unitRingType.
Implicit Types x y : R.
Implicit Types m n : int.
Local Coercion Posz : nat >-> int.

Lemma exprz_pmulzl x m n : 0 <= n -> (x *~ m) ^ n = x ^ n *~ (m ^ n).
Proof.
by elim: n=> [|n ihn|n _] // _; rewrite !exprSz ihn // mulrzAr mulrzAl -mulrzA.
Qed.

Lemma exprz_pintl m n (hn : 0 <= n) : m%:~R ^ n = (m ^ n)%:~R :> R.
Proof. by rewrite exprz_pmulzl // exp1rz. Qed.

Lemma exprzMzl x m n (ux : x \is a GRing.unit) (um : m%:~R \is a @GRing.unit R):
   (x *~ m) ^ n = (m%:~R ^ n) * x ^ n :> R.
Proof.
rewrite -[x *~ _]mulrzl exprMz_comm //.
by apply: commr_sym; apply: commr_int.
Qed.

Lemma expNrz x n : (- x) ^ n = (-1) ^ n * x ^ n :> R.
Proof.
case: n=> [] n; rewrite ?NegzE; first by apply: exprNn.
by rewrite -!exprz_inv !invrN invr1; apply: exprNn.
Qed.

Lemma unitr_n0expz x n :
  n != 0 -> (x ^ n \is a GRing.unit) = (x \is a GRing.unit).
Proof.
by case: n => *; rewrite ?NegzE -?exprz_inv ?unitrX_pos ?unitrV ?lt0n.
Qed.

Lemma intrV (n : int) :
  n \in [:: 0; 1; -1] -> n%:~R ^-1 = n%:~R :> R.
Proof.
by case: (intP n)=> // [|[]|[]] //; rewrite ?rmorphN ?invrN (invr0, invr1).
Qed.

Lemma rmorphXz (R' : unitRingType) (f : {rmorphism R -> R'}) n :
  {in GRing.unit, {morph f : x / x ^ n}}.
Proof. by case: n => n x Ux; rewrite ?rmorphV ?rpredX ?rmorphX. Qed.

End Exprz_Zint_UnitRing.

Section ExprzIdomain.

Variable R : idomainType.
Implicit Types x y : R.
Implicit Types m n : int.
Local Coercion Posz : nat >-> int.

Lemma expfz_eq0 x n : (x ^ n == 0) = (n != 0) && (x == 0).
Proof.
by case: n=> n; rewrite ?NegzE -?exprz_inv ?expf_eq0 ?lt0n ?invr_eq0.
Qed.

Lemma expfz_neq0 x n : x != 0 -> x ^ n != 0.
Proof. by move=> x_nz; rewrite expfz_eq0; apply/nandP; right. Qed.

Lemma exprzMl x y n (ux : x \is a GRing.unit) (uy : y \is a GRing.unit) :
  (x * y) ^ n = x ^ n * y ^ n.
Proof. by rewrite exprMz_comm //; apply: mulrC. Qed.

Lemma expfV (x : R) (i : int) : (x ^ i) ^-1 = (x ^-1) ^ i.
Proof. by rewrite invr_expz exprz_inv. Qed.

End ExprzIdomain.

Section ExprzField.

Variable F : fieldType.
Implicit Types x y : F.
Implicit Types m n : int.
Local Coercion Posz : nat >-> int.

Lemma expfzDr x m n : x != 0 -> x ^ (m + n) = x ^ m * x ^ n.
Proof. by move=> hx; rewrite exprzDr ?unitfE. Qed.

Lemma expfz_n0addr x m n : m + n != 0 -> x ^ (m + n) = x ^ m * x ^ n.
Proof.
have [-> hmn|nx0 _] := eqVneq x 0; last exact: expfzDr.
rewrite !exp0rz (negPf hmn).
case: (altP (m =P 0)) hmn=> [->|]; rewrite (mul0r, mul1r) //.
by rewrite add0r=> /negPf->.
Qed.

Lemma expfzMl x y n : (x * y) ^ n = x ^ n * y ^ n.
Proof.
have [->|/negPf n0] := eqVneq n 0; first by rewrite !expr0z mulr1.
case: (boolP ((x * y) == 0)); rewrite ?mulf_eq0.
  by case/orP=> /eqP->; rewrite ?(mul0r, mulr0, exp0rz, n0).
by case/norP=> x0 y0; rewrite exprzMl ?unitfE.
Qed.

Lemma fmorphXz (R : unitRingType) (f : {rmorphism F -> R}) n :
  {morph f : x / x ^ n}.
Proof. by case: n => n x; rewrite ?fmorphV rmorphX. Qed.

End ExprzField.

Section ExprzOrder.

Variable R : realFieldType.
Implicit Types x y : R.
Implicit Types m n : int.
Local Coercion Posz : nat >-> int.

(* ler and exprz *)
Lemma exprz_ge0 n x (hx : 0 <= x) : (0 <= x ^ n).
Proof. by case: n=> n; rewrite ?NegzE -?invr_expz ?invr_ge0 ?exprn_ge0. Qed.

Lemma exprz_gt0 n x (hx : 0 < x) : (0 < x ^ n).
Proof. by case: n=> n; rewrite ?NegzE -?invr_expz ?invr_gt0 ?exprn_gt0. Qed.

Definition exprz_gte0 := (exprz_ge0, exprz_gt0).

Lemma ler_wpiexpz2l x (x0 : 0 <= x) (x1 : x <= 1) :
  {in >= 0 &, {homo (exprz x) : x y /~ x <= y}}.
Proof.
move=> [] m [] n; rewrite -!topredE /= ?oppr_cp0 ?ltz_nat // => _ _.
by rewrite lez_nat -?exprnP=> /ler_wiexpn2l; apply.
Qed.

Lemma ler_wniexpz2l x (x0 : 0 <= x) (x1 : x <= 1) :
  {in < 0 &, {homo (exprz x) : x y /~ x <= y}}.
Proof.
move=> [] m [] n; rewrite ?NegzE -!topredE /= ?oppr_cp0 ?ltz_nat // => _ _.
rewrite ler_opp2 lez_nat -?invr_expz=> hmn; move: (x0).
rewrite le0r=> /orP [/eqP->|lx0]; first by rewrite !exp0rz invr0.
by rewrite lef_pinv -?topredE /= ?exprz_gt0 // ler_wiexpn2l.
Qed.

Fact ler_wpeexpz2l x (x1 : 1 <= x) :
  {in >= 0 &, {homo (exprz x) : x y / x <= y}}.
Proof.
move=> [] m [] n; rewrite -!topredE /= ?oppr_cp0 ?ltz_nat // => _ _.
by rewrite lez_nat -?exprnP=> /ler_weexpn2l; apply.
Qed.

Fact ler_wneexpz2l x (x1 : 1 <= x) :
  {in <= 0 &, {homo (exprz x) : x y / x <= y}}.
Proof.
move=> m n hm hn /= hmn.
rewrite -lef_pinv -?topredE /= ?exprz_gt0 ?(ltr_le_trans ltr01) //.
by rewrite !invr_expz ler_wpeexpz2l ?ler_opp2 -?topredE //= oppr_cp0.
Qed.

Lemma ler_weexpz2l x (x1 : 1 <= x) : {homo (exprz x) : x y / x <= y}.
Proof.
move=> m n /= hmn; case: (lerP 0 m)=> [|/ltrW] hm.
  by rewrite ler_wpeexpz2l // [_ \in _](ler_trans hm).
case: (lerP n 0)=> [|/ltrW] hn.
  by rewrite ler_wneexpz2l // [_ \in _](ler_trans hmn).
apply: (@ler_trans _ (x ^ 0)); first by rewrite ler_wneexpz2l.
by rewrite ler_wpeexpz2l.
Qed.

Lemma pexprz_eq1 x n (x0 : 0 <= x) : (x ^ n == 1) = ((n == 0) || (x == 1)).
Proof.
case: n=> n; rewrite ?NegzE -?exprz_inv ?oppr_eq0 pexprn_eq1 // ?invr_eq1 //.
by rewrite invr_ge0.
Qed.

Lemma ieexprIz x (x0 : 0 < x) (nx1 : x != 1) : injective (exprz x).
Proof.
apply: wlog_ltr=> // m n hmn; first by move=> hmn'; rewrite hmn.
move=> /(f_equal ( *%R^~ (x ^ (- n)))).
rewrite -!expfzDr ?gtr_eqF // subrr expr0z=> /eqP.
by rewrite pexprz_eq1 ?(ltrW x0) // (negPf nx1) subr_eq0 orbF=> /eqP.
Qed.

Lemma ler_piexpz2l x (x0 : 0 < x) (x1 : x < 1) :
  {in >= 0 &, {mono (exprz x) : x y /~ x <= y}}.
Proof.
apply: (nhomo_mono_in (nhomo_inj_in_lt _ _)).
  by move=> n m hn hm /=; apply: ieexprIz; rewrite // ltr_eqF.
by apply: ler_wpiexpz2l; rewrite ?ltrW.
Qed.

Lemma ltr_piexpz2l x (x0 : 0 < x) (x1 : x < 1) :
  {in >= 0 &, {mono (exprz x) : x y /~ x < y}}.
Proof. exact: (lerW_nmono_in (ler_piexpz2l _ _)). Qed.

Lemma ler_niexpz2l x (x0 : 0 < x) (x1 : x < 1) :
  {in < 0 &, {mono (exprz x) : x y /~ x <= y}}.
Proof.
apply: (nhomo_mono_in (nhomo_inj_in_lt _ _)).
  by move=> n m hn hm /=; apply: ieexprIz; rewrite // ltr_eqF.
by apply: ler_wniexpz2l; rewrite ?ltrW.
Qed.

Lemma ltr_niexpz2l x (x0 : 0 < x) (x1 : x < 1) :
  {in < 0 &, {mono (exprz x) : x y /~ x < y}}.
Proof. exact: (lerW_nmono_in (ler_niexpz2l _ _)). Qed.

Lemma ler_eexpz2l x (x1 : 1 < x) : {mono (exprz x) : x y / x <= y}.
Proof.
apply: (homo_mono (homo_inj_lt _ _)).
  by apply: ieexprIz; rewrite ?(ltr_trans ltr01) // gtr_eqF.
by apply: ler_weexpz2l; rewrite ?ltrW.
Qed.

Lemma ltr_eexpz2l x (x1 : 1 < x) : {mono (exprz x) : x y / x < y}.
Proof. exact: (lerW_mono (ler_eexpz2l _)). Qed.

Lemma ler_wpexpz2r n (hn : 0 <= n) :
{in >= 0 & , {homo ((@exprz R)^~ n) : x y / x <= y}}.
Proof. by case: n hn=> // n _; apply: ler_expn2r. Qed.

Lemma ler_wnexpz2r n (hn : n <= 0) :
{in > 0 & , {homo ((@exprz R)^~ n) : x y /~ x <= y}}.
Proof.
move=> x y /= hx hy hxy; rewrite -lef_pinv ?[_ \in _]exprz_gt0 //.
by rewrite !invr_expz ler_wpexpz2r  ?[_ \in _]ltrW // oppr_cp0.
Qed.

Lemma pexpIrz n (n0 : n != 0) : {in >= 0 &, injective ((@exprz R)^~ n)}.
Proof.
move=> x y; rewrite ![_ \in _]le0r=> /orP [/eqP-> _ /eqP|hx].
  by rewrite exp0rz ?(negPf n0) eq_sym expfz_eq0=> /andP [_ /eqP->].
case/orP=> [/eqP-> /eqP|hy].
  by rewrite exp0rz ?(negPf n0) expfz_eq0=> /andP [_ /eqP].
move=> /(f_equal ( *%R^~ (y ^ (- n)))) /eqP.
rewrite -expfzDr ?(gtr_eqF hy) // subrr expr0z -exprz_inv -expfzMl.
rewrite pexprz_eq1 ?(negPf n0) /= ?mulr_ge0 ?invr_ge0 ?ltrW //.
by rewrite (can2_eq (mulrVK _) (mulrK _)) ?unitfE ?(gtr_eqF hy) // mul1r=> /eqP.
Qed.

Lemma nexpIrz n (n0 : n != 0) : {in <= 0 &, injective ((@exprz R)^~ n)}.
Proof.
move=> x y; rewrite ![_ \in _]ler_eqVlt => /orP [/eqP -> _ /eqP|hx].
  by rewrite exp0rz ?(negPf n0) eq_sym expfz_eq0=> /andP [_ /eqP->].
case/orP=> [/eqP -> /eqP|hy].
  by rewrite exp0rz ?(negPf n0) expfz_eq0=> /andP [_ /eqP].
move=> /(f_equal ( *%R^~ (y ^ (- n)))) /eqP.
rewrite -expfzDr ?(ltr_eqF hy) // subrr expr0z -exprz_inv -expfzMl.
rewrite pexprz_eq1 ?(negPf n0) /= ?mulr_le0 ?invr_le0 ?ltrW //.
by rewrite (can2_eq (mulrVK _) (mulrK _)) ?unitfE ?(ltr_eqF hy) // mul1r=> /eqP.
Qed.

Lemma ler_pexpz2r n (hn : 0 < n) :
  {in >= 0 & , {mono ((@exprz R)^~ n) : x y / x <= y}}.
Proof.
apply: homo_mono_in (homo_inj_in_lt _ _).
  by move=> x y hx hy /=; apply: pexpIrz; rewrite // gtr_eqF.
by apply: ler_wpexpz2r; rewrite ltrW.
Qed.

Lemma ltr_pexpz2r n (hn : 0 < n) :
  {in >= 0 & , {mono ((@exprz R)^~ n) : x y / x < y}}.
Proof. exact: lerW_mono_in (ler_pexpz2r _). Qed.

Lemma ler_nexpz2r n (hn : n < 0) :
  {in > 0 & , {mono ((@exprz R)^~ n) : x y /~ x <= y}}.
Proof.
apply: nhomo_mono_in (nhomo_inj_in_lt _ _); last first.
  by apply: ler_wnexpz2r; rewrite ltrW.
by move=> x y hx hy /=; apply: pexpIrz; rewrite ?[_ \in _]ltrW ?ltr_eqF.
Qed.

Lemma ltr_nexpz2r n (hn : n < 0) :
  {in > 0 & , {mono ((@exprz R)^~ n) : x y /~ x < y}}.
Proof. exact: lerW_nmono_in (ler_nexpz2r _). Qed.

Lemma eqr_expz2 n x y : n != 0 -> 0 <= x -> 0 <= y ->
  (x ^ n == y ^ n) = (x == y).
Proof. by  move=> *; rewrite (inj_in_eq (pexpIrz _)). Qed.

End ExprzOrder.

Local Notation sgr := Num.sg.

Section Sgz.

Variable R : numDomainType.
Implicit Types x y z : R.
Implicit Types m n p : int.
Local Coercion Posz : nat >-> int.

Definition sgz x : int := if x == 0 then 0 else if x < 0 then -1 else 1.

Lemma sgz_def x : sgz x = (-1) ^+ (x < 0)%R *+ (x != 0).
Proof. by rewrite /sgz; case: (_ == _); case: (_ < _). Qed.

Lemma sgrEz x : sgr x = (sgz x)%:~R. Proof. by rewrite !(fun_if intr). Qed.

Lemma gtr0_sgz x : 0 < x -> sgz x = 1.
Proof. by move=> x_gt0; rewrite /sgz ltr_neqAle andbC eqr_le ltr_geF //. Qed.

Lemma ltr0_sgz x : x < 0 -> sgz x = -1.
Proof. by move=> x_lt0; rewrite /sgz eq_sym eqr_le x_lt0 ltr_geF. Qed.

Lemma sgz0 : sgz (0 : R) = 0. Proof. by rewrite /sgz eqxx. Qed.
Lemma sgz1 : sgz (1 : R) = 1. Proof. by rewrite gtr0_sgz // ltr01. Qed.
Lemma sgzN1 : sgz (-1 : R) = -1. Proof. by rewrite ltr0_sgz // ltrN10. Qed.

Definition sgzE := (sgz0, sgz1, sgzN1).

Lemma sgz_sgr x : sgz (sgr x) = sgz x.
Proof. by rewrite !(fun_if sgz) !sgzE. Qed.

Lemma normr_sgz x : `|sgz x| = (x != 0).
Proof. by rewrite sgz_def -mulr_natr normrMsign normr_nat natz. Qed.

Lemma normr_sg x : `|sgr x| = (x != 0)%:~R.
Proof. by rewrite sgr_def -mulr_natr normrMsign normr_nat. Qed.

End Sgz.

Section MoreSgz.

Variable R : numDomainType.

Lemma sgz_int m : sgz (m%:~R : R) = sgz m.
Proof. by rewrite /sgz intr_eq0 ltrz0. Qed.

Lemma sgrz (n : int) : sgr n = sgz n. Proof. by rewrite sgrEz intz. Qed.

Lemma intr_sg m : (sgr m)%:~R = sgr (m%:~R) :> R.
Proof. by rewrite sgrz -sgz_int -sgrEz. Qed.

Lemma sgz_id (x : R) : sgz (sgz x) = sgz x.
Proof. by rewrite !(fun_if (@sgz _)). Qed.

End MoreSgz.

Section SgzReal.

Variable R : realDomainType.
Implicit Types x y z : R.
Implicit Types m n p : int.
Local Coercion Posz : nat >-> int.

Lemma sgz_cp0 x :
  ((sgz x == 1) = (0 < x)) *
  ((sgz x == -1) = (x < 0)) *
  ((sgz x == 0) = (x == 0)).
Proof. by rewrite /sgz; case: ltrgtP. Qed.

CoInductive sgz_val x : bool -> bool -> bool -> bool -> bool -> bool
  -> bool -> bool -> bool -> bool -> bool -> bool
  -> bool -> bool -> bool -> bool -> bool -> bool
  -> R -> R -> int -> Set :=
  | SgzNull of x = 0 : sgz_val x true true true true false false
    true false false true false false true false false true false false 0 0 0
  | SgzPos of x > 0 : sgz_val x false false true false false true
    false false true false false true false false true false false true x 1 1
  | SgzNeg of x < 0 : sgz_val x false true false false true false
    false true false false true false false true false false true false (-x) (-1) (-1).

Lemma sgzP x :
  sgz_val x (0 == x) (x <= 0) (0 <= x) (x == 0) (x < 0) (0 < x)
  (0 == sgr x) (-1 == sgr x) (1 == sgr x)
  (sgr x == 0)  (sgr x == -1) (sgr x == 1)
  (0 == sgz x) (-1 == sgz x) (1 == sgz x)
  (sgz x == 0)  (sgz x == -1) (sgz x == 1) `|x| (sgr x) (sgz x).
Proof.
rewrite ![_ == sgz _]eq_sym ![_ == sgr _]eq_sym !sgr_cp0 !sgz_cp0.
by rewrite /sgr /sgz !lerNgt; case: ltrgt0P; constructor.
Qed.

Lemma sgzN x : sgz (- x) = - sgz x.
Proof. by rewrite /sgz oppr_eq0 oppr_lt0; case: ltrgtP. Qed.

Lemma mulz_sg x : sgz x * sgz x = (x != 0)%:~R.
Proof. by case: sgzP; rewrite ?(mulr0, mulr1, mulrNN). Qed.

Lemma mulz_sg_eq1 x y : (sgz x * sgz y == 1) = (x != 0) && (sgz x == sgz y).
Proof.
do 2?case: sgzP=> _; rewrite ?(mulr0, mulr1, mulrN1, opprK, oppr0, eqxx);
  by rewrite ?[0 == 1]eq_sym ?oner_eq0 //= eqr_oppLR oppr0 oner_eq0.
Qed.

Lemma mulz_sg_eqN1 x y : (sgz x * sgz y == -1) = (x != 0) && (sgz x == - sgz y).
Proof. by rewrite -eqr_oppLR -mulrN -sgzN mulz_sg_eq1. Qed.

(* Lemma muls_eqA x y z : sgr x != 0 -> *)
(*   (sgr y * sgr z == sgr x) = ((sgr y * sgr x == sgr z) && (sgr z != 0)). *)
(* Proof. by do 3!case: sgrP=> _. Qed. *)

Lemma sgzM x y : sgz (x * y) = sgz x  * sgz y.
Proof.
case: (sgzP x)=> hx; first by rewrite hx ?mul0r sgz0.
  case: (sgzP y)=> hy; first by rewrite hy !mulr0 sgz0.
    by apply/eqP; rewrite mul1r sgz_cp0 pmulr_rgt0.
  by apply/eqP; rewrite mul1r sgz_cp0 nmulr_llt0.
case: (sgzP y)=> hy; first by rewrite hy !mulr0 sgz0.
  by apply/eqP; rewrite mulr1 sgz_cp0 nmulr_rlt0.
by apply/eqP; rewrite mulN1r opprK sgz_cp0 nmulr_rgt0.
Qed.

Lemma sgzX (n : nat) x : sgz (x ^+ n) = (sgz x) ^+ n.
Proof. by elim: n => [|n IHn]; rewrite ?sgz1 // !exprS sgzM IHn. Qed.

Lemma sgz_eq0 x : (sgz x == 0) = (x == 0).
Proof. by rewrite sgz_cp0. Qed.

Lemma sgz_odd (n : nat) x : x != 0 -> (sgz x) ^+ n = (sgz x) ^+ (odd n).
Proof. by case: sgzP => //=; rewrite ?expr1n // signr_odd. Qed.

Lemma sgz_gt0 x : (sgz x > 0) = (x > 0).
Proof. by case: sgzP. Qed.

Lemma sgz_lt0 x : (sgz x < 0) = (x < 0).
Proof. by case: sgzP. Qed.

Lemma sgz_ge0 x : (sgz x >= 0) = (x >= 0).
Proof. by case: sgzP. Qed.

Lemma sgz_le0 x : (sgz x <= 0) = (x <= 0).
Proof. by case: sgzP. Qed.

Lemma sgz_smul x y : sgz (y *~ (sgz x)) = (sgz x) * (sgz y).
Proof. by rewrite -mulrzl sgzM -sgrEz sgz_sgr. Qed.

Lemma sgrMz m x : sgr (x *~ m) = sgr x *~ sgr m.
Proof. by rewrite -mulrzr sgrM -intr_sg mulrzr. Qed.

End SgzReal.

Lemma sgz_eq (R R' : realDomainType) (x : R) (y : R') :
  (sgz x == sgz y) = ((x == 0) == (y == 0)) && ((0 < x) == (0 < y)).
Proof. by do 2!case: sgzP. Qed.

Lemma intr_sign (R : ringType) s : ((-1) ^+ s)%:~R = (-1) ^+ s :> R.
Proof. exact: rmorph_sign. Qed.

Section Absz.

Implicit Types m n p : int.
Open Scope nat_scope.
Local Coercion Posz : nat >-> int.

Lemma absz_nat (n : nat) : `|n| = n. Proof. by []. Qed.

Lemma abszE (m : int) : `|m| = `|m|%R :> int. Proof. by []. Qed.

Lemma absz0 : `|0%R| = 0. Proof. by []. Qed.

Lemma abszN m : `|- m| = `|m|. Proof. by case: (normrN m). Qed.

Lemma absz_eq0 m : (`|m| == 0) = (m == 0%R). Proof. by case: (intP m). Qed.

Lemma absz_gt0 m : (`|m| > 0) = (m != 0%R). Proof. by case: (intP m). Qed.

Lemma absz1 : `|1%R| = 1. Proof. by []. Qed.

Lemma abszN1 : `|-1%R| = 1. Proof. by []. Qed.

Lemma absz_id m : `|(`|m|)| = `|m|. Proof. by []. Qed.

Lemma abszM m1 m2 : `|(m1 * m2)%R| = `|m1| * `|m2|.
Proof. by case: m1 m2 => [[|m1]|m1] [[|m2]|m2]; rewrite //= mulnS mulnC. Qed.

Lemma abszX (n : nat) m : `|m ^+ n| = `|m| ^ n.
Proof. by elim: n => // n ihn; rewrite exprS expnS abszM ihn. Qed.

Lemma absz_sg m : `|sgr m| = (m != 0%R). Proof. by case: (intP m). Qed.

Lemma gez0_abs m : (0 <= m)%R -> `|m| = m :> int.
Proof. by case: (intP m). Qed.

Lemma gtz0_abs m : (0 < m)%R -> `|m| = m :> int.
Proof. by case: (intP m). Qed.

Lemma lez0_abs m : (m <= 0)%R -> `|m| = - m :> int.
Proof. by case: (intP m). Qed.

Lemma ltz0_abs m : (m < 0)%R -> `|m| = - m :> int.
Proof. by case: (intP m). Qed.

Lemma absz_sign s : `|(-1) ^+ s| = 1.
Proof. by rewrite abszX exp1n. Qed.

Lemma abszMsign s m : `|((-1) ^+ s * m)%R| = `|m|.
Proof. by rewrite abszM absz_sign mul1n. Qed.

Lemma mulz_sign_abs m : ((-1) ^+ (m < 0)%R * `|m|%:Z)%R = m.
Proof. by rewrite abszE mulr_sign_norm. Qed.

Lemma mulz_Nsign_abs m : ((-1) ^+ (0 < m)%R * `|m|%:Z)%R = - m.
Proof. by rewrite abszE mulr_Nsign_norm. Qed.

Lemma intEsign  m : m = ((-1) ^+ (m < 0)%R * `|m|%:Z)%R.
Proof. exact: numEsign. Qed.

Lemma abszEsign m : `|m|%:Z = ((-1) ^+ (m < 0)%R * m)%R.
Proof. exact: normrEsign. Qed.

Lemma intEsg m : m = (sgz m * `|m|%:Z)%R.
Proof. by rewrite -sgrz -numEsg. Qed.

Lemma abszEsg m : (`|m|%:Z = sgz m * m)%R.
Proof. by rewrite -sgrz -normrEsg. Qed.

End Absz.

Module Export IntDist.

Notation "m - n" :=
  (@GRing.add int_ZmodType m%N (@GRing.opp int_ZmodType n%N)) : distn_scope.
Arguments Scope absz [distn_scope].
Notation "`| m |" := (absz m) : nat_scope.
Coercion Posz : nat >-> int.

Section Distn.

Open Scope nat_scope.
Implicit Type m : int.
Implicit Types n d : nat.

Lemma distnC m1 m2 : `|m1 - m2| = `|m2 - m1|.
Proof. by rewrite -opprB abszN. Qed.

Lemma distnDl d n1 n2 : `|d + n1 - (d + n2)| = `|n1 - n2|.
Proof. by rewrite !PoszD opprD addrCA -addrA addKr. Qed.

Lemma distnDr d n1 n2 : `|n1 + d - (n2 + d)| = `|n1 - n2|.
Proof. by rewrite -!(addnC d) distnDl. Qed.

Lemma distnEr n1 n2 : n1 <= n2 -> `|n1 - n2| = n2 - n1.
Proof. by move/subnK=> {1}<-; rewrite distnC PoszD addrK absz_nat. Qed.

Lemma distnEl n1 n2 : n2 <= n1 -> `|n1 - n2| = n1 - n2.
Proof. by move/distnEr <-; rewrite distnC. Qed.

Lemma distn0 n : `|n - 0| = n.
Proof. by rewrite subr0 absz_nat. Qed.

Lemma dist0n n : `|0 - n| = n.
Proof. by rewrite distnC distn0. Qed.

Lemma distnn m : `|m - m| = 0.
Proof. by rewrite subrr. Qed.

Lemma distn_eq0 n1 n2 : (`|n1 - n2| == 0) = (n1 == n2).
Proof. by rewrite absz_eq0 subr_eq0. Qed.

Lemma distnS n : `|n - n.+1| = 1.
Proof. exact: distnDr n 0 1. Qed.

Lemma distSn n : `|n.+1 - n| = 1.
Proof. exact: distnDr n 1 0. Qed.

Lemma distn_eq1 n1 n2 :
  (`|n1 - n2| == 1) = (if n1 < n2 then n1.+1 == n2 else n1 == n2.+1).
Proof.
case: ltnP => [lt_n12 | le_n21].
  by rewrite eq_sym -(eqn_add2r n1) distnEr ?subnK // ltnW.
by rewrite -(eqn_add2r n2) distnEl ?subnK.
Qed.

Lemma leq_add_dist  m1 m2 m3 : `|m1 - m3| <= `|m1 - m2| + `|m2 - m3|.
Proof. by rewrite -lez_nat PoszD !abszE ler_dist_add. Qed.

(* Most of this proof generalizes to all real-ordered rings. *)
Lemma leqif_add_distz m1 m2 m3 :
  `|m1 - m3| <= `|m1 - m2| + `|m2 - m3|
             ?= iff (m1 <= m2 <= m3)%R || (m3 <= m2 <= m1)%R.
Proof.
apply/leqifP; rewrite -ltz_nat -eqz_nat PoszD !abszE; apply/lerifP.
wlog le_m31 : m1 m3 / (m3 <= m1)%R.
  move=> IH; case/orP: (ler_total m1 m3) => /IH //.
  by rewrite (addrC `|_|)%R orbC !(distrC m1) !(distrC m3).
rewrite ger0_norm ?subr_ge0 // orb_idl => [|/andP[le_m12 le_m23]]; last first.
  by have /eqP->: m2 == m3; rewrite ?lerr // eqr_le le_m23 (ler_trans le_m31).
rewrite -{1}(subrK m2 m1) -addrA -subr_ge0 andbC -subr_ge0.
by apply: lerif_add; apply/real_lerif_norm/num_real.
Qed.

Lemma leqif_add_dist n1 n2 n3 :
  `|n1 - n3| <= `|n1 - n2| + `|n2 - n3|
             ?= iff (n1 <= n2 <= n3) || (n3 <= n2 <= n1).
Proof. exact: leqif_add_distz. Qed.

Lemma sqrn_dist n1 n2 : `|n1 - n2| ^ 2 + 2 * (n1 * n2) = n1 ^ 2 + n2 ^ 2.
Proof.
wlog le_n21: n1 n2 / n2 <= n1.
  move=> IH; case/orP: (leq_total n2 n1) => /IH //.
  by rewrite (addnC (n2 ^ 2)) (mulnC n2) distnC.
by rewrite distnEl ?sqrn_sub ?subnK ?nat_Cauchy.
Qed.

End Distn.

End IntDist.

Section NormInt.

Variable R : numDomainType.

Lemma intr_norm m : `|m|%:~R = `|m%:~R| :> R.
Proof. by rewrite {2}[m]intEsign rmorphMsign normrMsign abszE normr_nat. Qed.

Lemma normrMz m (x : R) : `|x *~ m| = `|x| *~ `|m|.
Proof. by rewrite -mulrzl normrM -intr_norm mulrzl. Qed.

Lemma expN1r (i : int) : (-1 : R) ^ i = (-1) ^+ `|i|.
Proof.
case: i => n; first by rewrite exprnP absz_nat.
by rewrite NegzE abszN  absz_nat -invr_expz expfV invrN1.
Qed.

End NormInt.

Section PolyZintRing.

Variable R : ringType.
Implicit Types x y z: R.
Implicit Types m n : int.
Implicit Types i j k : nat.
Implicit Types p q r : {poly R}.

Lemma coefMrz : forall p n i, (p *~ n)`_i = (p`_i *~ n).
Proof. by move=> p [] n i; rewrite ?NegzE (coefMNn, coefMn). Qed.

Lemma polyC_mulrz : forall n, {morph (@polyC R) : c / c *~ n}.
Proof.
move=> [] n c; rewrite ?NegzE -?pmulrn ?polyC_muln //.
by rewrite polyC_opp mulrNz polyC_muln nmulrn.
Qed.

Lemma hornerMz : forall n (p : {poly R}) x, (p *~ n).[x] = p.[x] *~ n.
Proof. by case=> *; rewrite ?NegzE ?mulNzr ?(hornerN, hornerMn). Qed.

Lemma horner_int : forall n x, (n%:~R : {poly R}).[x] = n%:~R.
Proof. by move=> n x; rewrite hornerMz hornerC. Qed.

Lemma derivMz : forall n p, (p *~ n)^`() = p^`() *~ n.
Proof. by move=> [] n p; rewrite ?NegzE -?pmulrn (derivMn, derivMNn). Qed.

End PolyZintRing.

Section PolyZintOIdom.

Variable R : realDomainType.

Lemma mulpz (p : {poly R}) (n : int) : p *~ n = n%:~R *: p.
Proof. by rewrite -[p *~ n]mulrzl -mul_polyC polyC_mulrz polyC1. Qed.

End PolyZintOIdom.

Section ZnatPred.

Definition Znat := [qualify a n : int | 0 <= n].
Fact Znat_key : pred_key Znat. by []. Qed.
Canonical Znat_keyd := KeyedQualifier Znat_key.

Lemma Znat_def n : (n \is a Znat) = (0 <= n). Proof. by []. Qed.

Lemma Znat_semiring_closed : semiring_closed Znat.
Proof. by do 2?split => //; [apply: addr_ge0 | apply: mulr_ge0]. Qed.
Canonical Znat_addrPred := AddrPred Znat_semiring_closed.
Canonical Znat_mulrPred := MulrPred Znat_semiring_closed.
Canonical Znat_semiringPred := SemiringPred Znat_semiring_closed.

Lemma ZnatP (m : int) : reflect (exists n : nat, m = n) (m \is a Znat).
Proof. by apply: (iffP idP) => [|[n -> //]]; case: m => // n; exists n. Qed.

End ZnatPred.

Section rpred.

Lemma rpredMz M S (addS : @zmodPred M S) (kS : keyed_pred addS) m :
  {in kS, forall u, u *~ m \in kS}.
Proof. by case: m => n u Su; rewrite ?rpredN ?rpredMn. Qed.

Lemma rpred_int R S (ringS : @subringPred R S) (kS : keyed_pred ringS) m :
  m%:~R \in kS.
Proof. by rewrite rpredMz ?rpred1. Qed.

Lemma rpredZint (R : ringType) (M : lmodType R) S
                 (addS : @zmodPred M S) (kS : keyed_pred addS) m :
  {in kS, forall u, m%:~R *: u \in kS}.
Proof. by move=> u Su; rewrite /= scaler_int rpredMz. Qed.

Lemma rpredXz R S (divS : @divrPred R S) (kS : keyed_pred divS) m :
  {in kS, forall x, x ^ m \in kS}.
Proof. by case: m => n x Sx; rewrite ?rpredV rpredX. Qed.

Lemma rpredXsign R S (divS : @divrPred R S) (kS : keyed_pred divS) n x :
  (x ^ ((-1) ^+ n) \in kS) = (x \in kS).
Proof. by rewrite -signr_odd; case: (odd n); rewrite ?rpredV. Qed.

End rpred.