Library mathcomp.algebra.ssrnum

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
Distributed under the terms of CeCILL-B. *)
From HB Require Import structures.
From mathcomp Require Import ssreflect ssrfun ssrbool ssrnotations eqtype.
From mathcomp Require Import ssrnat seq choice ssrAC div fintype path bigop.
From mathcomp Require Import order finset fingroup ssralg poly.

Number structures NB: See CONTRIBUTING.md for an introduction to HB concepts and commands. This file defines some classes to manipulate number structures, i.e, structures with an order and a norm. To use this file, insert "Import Num.Theory." before your scripts. You can also "Import Num.Def." to enjoy shorter notations (e.g., minr instead of Num.min, lerif instead of Num.leif, etc.). This file defines the following number structures: porderZmodType == join of Order.POrder and GRing.Zmodule The HB class is called POrderedZmodule. semiNormedZmodType == Zmodule with a semi-norm The HB class is called SemiNormedZmodule. normedZmodType == Zmodule with a norm The HB class is called NormedZmodule. numDomainType == Integral domain with an order and a norm The HB class is called NumDomain. numFieldType == Field with an order and a norm The HB class is called NumField. numClosedFieldType == Partially ordered Closed Field with conjugation The HB class is called ClosedField. realDomainType == Num domain where all elements are positive or negative The HB class is called RealDomain. realFieldType == Num Field where all elements are positive or negative The HB class is called RealField. rcfType == A Real Field with the real closed axiom The HB class is called RealClosedField. The ordering symbols and notations (<, <=, >, >=, _ <= _ ?= iff _, _ < _ ?<= if _, >=<, and ><) and lattice operations (meet and join) defined in order.v are redefined for the ring_display in the ring_scope (%R). 0-ary ordering symbols for the ring_display have the suffix "%R", e.g., <%R. All the other ordering notations are the same as order.v. Over these structures, we have the following operations: `|x| == norm of x Num.sg x == sign of x: equal to 0 iff x = 0, to 1 iff x > 0, and to -1 in all other cases (including x < 0) x \is a Num.pos <=> x is positive (:= x > 0) x \is a Num.neg <=> x is negative (:= x < 0) x \is a Num.nneg <=> x is positive or 0 (:= x >= 0) x \is a Num.npos <=> x is negative or 0 (:= x <= 0) x \is a Num.real <=> x is real (:= x >= 0 or x < 0) Num.sqrt x == in a real-closed field, a positive square root of x if x >= 0, or 0 otherwise For numeric algebraically closed fields we provide the generic definitions 'i == the imaginary number (:= sqrtC (-1)) 'Re z == the real component of z 'Im z == the imaginary component of z z^* == the complex conjugate of z (:= conjC z) sqrtC z == a nonnegative square root of z, i.e., 0 <= sqrt x if 0 <= x n.-root z == more generally, for n > 0, an nth root of z, chosen with a minimal non-negative argument for n > 1 (i.e., with a maximal real part subject to a nonnegative imaginary part) Note that n.-root (-1) is a primitive 2nth root of unity, an thus not equal to -1 for n odd > 1 (this will be shown in file cyclotomic.v).

list of prefixes : p : positive n : negative sp : strictly positive sn : strictly negative i : interior = in [0, 1] or ]0, 1[ e : exterior = in [1, +oo[ or ]1; +oo[ w : non strict (weak) monotony

Pdeg2.NumClosed : theory of the degree 2 polynomials on NumClosedField. Pdeg2.NumClosedMonic : theory of Pdeg2.NumClosed specialized to monic polynomials. Pdeg2.Real : theory of the degree 2 polynomials on RealField and rcfType. Pdeg2.RealMonic : theory of Pdeg2.Real specialized to monic polynomials.

Set Implicit Arguments.

Reserved Notation "n .-root" (at level 2, format "n .-root").
Reserved Notation "'i" (at level 0).
Reserved Notation "'Re z" (at level 10, z at level 8).
Reserved Notation "'Im z" (at level 10, z at level 8).

Local Open Scope order_scope.
Local Open Scope ring_scope.
Import Order.TTheory GRing.Theory.

Fact ring_display : Order.disp_t.

Module Num.

#[short (type ="porderZmodType")]
HB.structure Definition POrderedZmodule :=
  { R of Order.isPOrder ring_display R & GRing.Zmodule R }.

#[short (type ="semiNormedZmodType")]
HB.structure Definition SemiNormedZmodule (R : porderZmodType ) :=
  { M of Zmodule_isSemiNormed R M & GRing.Zmodule M }.

#[short (type ="normedZmodType")]
HB.structure Definition NormedZmodule (R : porderZmodType ) :=
  { M of SemiNormedZmodule_isPositiveDefinite R M & SemiNormedZmodule R M }.
Arguments norm {R M} x : rename.

Module NormedZmoduleExports.
Bind Scope ring_scope with NormedZmodule.sort.

Notation " [ 'normedZmodType' R 'of' T 'for' cT ]" := (@clone _ (Phant R) T cT _ idfun) (at level 0, format " [ 'normedZmodType' R 'of' T 'for' cT ]") : form_scope. Notation " [ 'normedZmodType' R 'of' T ]" := (@clone _ (Phant R) T _ _ id) (at level 0, format " [ 'normedZmodType' R 'of' T ]") : form_scope.

End NormedZmoduleExports.

#[short (type ="numDomainType")]
HB.structure Definition NumDomain := { R of
     GRing.IntegralDomain R &
     POrderedZmodule R &
     NormedZmodule (POrderedZmodule.clone R _) R &
     isNumRing R
  }.
Arguments addr_gt0 {_} [x y] : rename.
Arguments ger_leVge {_} [x y] : rename.

TODO: make isNumDomain depend on intermediate structures TODO: make isNumDomain.sort canonically a NumDomain

Module NumDomainExports.
Bind Scope ring_scope with NumDomain.sort.
End NumDomainExports.

Module Import Def.

Notation normr := norm.
Notation "`| x |" := (norm x) : ring_scope.

Notation ler := (@Order.le ring_display _) (only parsing).
Notation "@ 'ler' R" := (@Order.le ring_display R)
  (at level 10, R at level 8, only parsing) : function_scope.
Notation ltr := (@Order.lt ring_display _) (only parsing).
Notation "@ 'ltr' R" := (@Order.lt ring_display R)
  (at level 10, R at level 8, only parsing) : function_scope.
Notation ger := (@Order.ge ring_display _) (only parsing).
Notation "@ 'ger' R" := (@Order.ge ring_display R)
  (at level 10, R at level 8, only parsing) : function_scope.
Notation gtr := (@Order.gt ring_display _) (only parsing).
Notation "@ 'gtr' R" := (@Order.gt ring_display R)
  (at level 10, R at level 8, only parsing) : function_scope.
Notation lerif := (@Order.leif ring_display _) (only parsing).
Notation "@ 'lerif' R" := (@Order.leif ring_display R)
  (at level 10, R at level 8, only parsing) : function_scope.
Notation lterif := (@Order.lteif ring_display _) (only parsing).
Notation "@ 'lteif' R" := (@Order.lteif ring_display R)
  (at level 10, R at level 8, only parsing) : function_scope.
Notation comparabler := (@Order.comparable ring_display _) (only parsing).
Notation "@ 'comparabler' R" := (@Order.comparable ring_display R)
  (at level 10, R at level 8, only parsing) : function_scope.
Notation maxr := (@Order.max ring_display _).
Notation "@ 'maxr' R" := (@Order.max ring_display R)
    (at level 10, R at level 8, only parsing) : function_scope.
Notation minr := (@Order.min ring_display _).
Notation "@ 'minr' R" := (@Order.min ring_display R)
  (at level 10, R at level 8, only parsing) : function_scope.

Section NumDomainDef.
Context {R : numDomainType}.

Definition sgr (x : R) : R := if x == 0 then 0 else if x < 0 then -1 else 1.
Definition Rpos_pred := fun x : R ⇒ 0 < x.
Definition Rpos : qualifier 0 R := [qualify x | Rpos_pred x ].
Definition Rneg_pred := fun x : R ⇒ x < 0.
Definition Rneg : qualifier 0 R := [qualify x : R | Rneg_pred x ].
Definition Rnneg_pred := fun x : R ⇒ 0 ≤ x.
Definition Rnneg : qualifier 0 R := [qualify x : R | Rnneg_pred x ].
Definition Rnpos_pred := fun x : R ⇒ x ≤ 0.
Definition Rnpos : qualifier 0 R := [qualify x : R | Rnpos_pred x ].
Definition Rreal_pred := fun x : R ⇒ (0 ≤ x ) || (x ≤ 0).
Definition Rreal : qualifier 0 R := [qualify x : R | Rreal_pred x ].

End NumDomainDef.

End Def.

Arguments Rpos_pred _ _ /.
Arguments Rneg_pred _ _ /.
Arguments Rnneg_pred _ _ /.
Arguments Rreal_pred _ _ /.

Shorter qualified names, when Num.Def is not imported.

Notation le := ler (only parsing).
Notation lt := ltr (only parsing).
Notation ge := ger (only parsing).
Notation gt := gtr (only parsing).
Notation leif := lerif (only parsing).
Notation lteif := lterif (only parsing).
Notation comparable := comparabler (only parsing).
Notation sg := sgr.
Notation max := maxr.
Notation min := minr.
Notation pos := Rpos.
Notation neg := Rneg.
Notation nneg := Rnneg.
Notation npos := Rnpos.
Notation real := Rreal.

(Exported) symbolic syntax.

Module Import Syntax.
Import Def.

Notation "`| x |" := (norm x) : ring_scope.

Notation "<=%R" := le : function_scope.
Notation ">=%R" := ge : function_scope.
Notation "<%R" := lt : function_scope.
Notation ">%R" := gt : function_scope.
Notation "<?=%R" := leif : function_scope.
Notation "<?<=%R" := lteif : function_scope.
Notation ">=<%R" := comparable : function_scope.
Notation "><%R" := (fun x y ⇒ ~~ (comparable x y )) : function_scope.

Notation "<= y" := (ge y) : ring_scope.
Notation "<= y :> T" := (≤ (y : T)) (only parsing) : ring_scope.
Notation ">= y" := (le y) : ring_scope.
Notation ">= y :> T" := (≥ (y : T)) (only parsing) : ring_scope.

Notation "< y" := (gt y) : ring_scope.
Notation "< y :> T" := (< (y : T)) (only parsing) : ring_scope.
Notation "> y" := (lt y) : ring_scope.
Notation "> y :> T" := (> (y : T)) (only parsing) : ring_scope.

Notation "x <= y" := (le x y) : ring_scope.
Notation "x <= y :> T" := ((x : T) ≤ (y : T)) (only parsing) : ring_scope.
Notation "x >= y" := (y ≤ x) (only parsing) : ring_scope.
Notation "x >= y :> T" := ((x : T) ≥ (y : T)) (only parsing) : ring_scope.

Notation "x < y" := (lt x y) : ring_scope.
Notation "x < y :> T" := ((x : T) < (y : T)) (only parsing) : ring_scope.
Notation "x > y" := (y < x) (only parsing) : ring_scope.
Notation "x > y :> T" := ((x : T) > (y : T)) (only parsing) : ring_scope.

Notation "x <= y <= z" := ((x ≤ y) && (y ≤ z)) : ring_scope.
Notation "x < y <= z" := ((x < y) && (y ≤ z)) : ring_scope.
Notation "x <= y < z" := ((x ≤ y) && (y < z)) : ring_scope.
Notation "x < y < z" := ((x < y) && (y < z)) : ring_scope.

Notation "x <= y ?= 'iff' C" := (lerif x y C) : ring_scope.
Notation "x <= y ?= 'iff' C :> R" := ((x : R) ≤ (y : R) ?= iff C)
  (only parsing) : ring_scope.

Notation "x < y ?<= 'if' C" := (lterif x y C) : ring_scope.
Notation "x < y ?<= 'if' C :> R" := ((x : R) < (y : R) ?<= if C)
  (only parsing) : ring_scope.

Notation ">=< y" := [pred x | comparable x y] : ring_scope.
Notation ">=< y :> T" := (>=< (y : T)) (only parsing) : ring_scope.
Notation "x >=< y" := (comparable x y) : ring_scope.

Notation ">< y" := [pred x | ~~ comparable x y] : ring_scope.
Notation ">< y :> T" := (>< (y : T)) (only parsing) : ring_scope.
Notation "x >< y" := (~~ (comparable x y)) : ring_scope.

Export Order.POCoercions.

End Syntax.

Section ExtensionAxioms.

Variable R : numDomainType.

Definition real_axiom : Prop := ∀ x : R, x \is real.

Definition archimedean_axiom : Prop := ∀ x : R, ∃ ub , `|x | < ub %:R.

Definition real_closed_axiom : Prop :=
  ∀ (p : {poly R }) (a b : R),
    a ≤ b → p .[a ] ≤ 0 ≤ p .[b ] → exists2 x , a ≤ x ≤ b & root p x.

End ExtensionAxioms.

The rest of the numbers interface hierarchy.

#[short (type ="numFieldType")]
HB.structure Definition NumField := { R of GRing.UnitRing_isField R &
     GRing.IntegralDomain R &
     POrderedZmodule R &
     NormedZmodule (POrderedZmodule.clone R _) R &
     isNumRing R }.

Module NumFieldExports.
Bind Scope ring_scope with NumField.sort.
End NumFieldExports.

#[short (type ="numClosedFieldType")]
HB.structure Definition ClosedField :=
  { R of NumField_isImaginary R & GRing.ClosedField R & NumField R }.

Module ClosedFieldExports.
Bind Scope ring_scope with ClosedField.sort.
End ClosedFieldExports.

#[short (type ="realDomainType")]
HB.structure Definition RealDomain :=
  { R of Order.Total ring_display R & NumDomain R }.

Module RealDomainExports.
Bind Scope ring_scope with RealDomain.sort.
End RealDomainExports.

#[short (type ="realFieldType")]
HB.structure Definition RealField :=
  { R of Order.Total ring_display R & NumField R }.

Module RealFieldExports.
Bind Scope ring_scope with RealField.sort.
End RealFieldExports.

#[short (type ="rcfType")]
HB.structure Definition RealClosedField :=
  { R of RealField_isClosed R & RealField R }.

Module RealClosedFieldExports.
Bind Scope ring_scope with RealClosedField.sort.
End RealClosedFieldExports.

The elementary theory needed to support the definition of the derived operations for the extensions described above.

Module Import Internals.

Section NumDomain.
Variable R : numDomainType.
Implicit Types x y : R.

Basic consequences (just enough to get predicate closure properties).

Lemma ger0_def x : (0 ≤ x ) = (`|x | == x ).

Lemma subr_ge0 x y : (0 ≤ x - y ) = (y ≤ x ).

Lemma oppr_ge0 x : (0 ≤ - x ) = (x ≤ 0).

Lemma ler01 : 0 ≤ 1 :> R.

Lemma ltr01 : 0 < 1 :> R.

Lemma le0r x : (0 ≤ x ) = (x == 0) || (0 < x ).

Lemma addr_ge0 x y : 0 ≤ x → 0 ≤ y → 0 ≤ x + y.

Lemma pmulr_rgt0 x y : 0 < x → (0 < x × y ) = (0 < y ).

Closure properties of the real predicates.

Lemma posrE x : (x \is pos ) = (0 < x ).
Lemma nnegrE x : (x \is nneg ) = (0 ≤ x ).
Lemma realE x : (x \is real ) = (0 ≤ x ) || (x ≤ 0).

Fact pos_divr_closed : divr_closed (@pos R).
#[export ]
HB.instance Definition _ := GRing.isDivClosed.Build R Rpos_pred
  pos_divr_closed.

Fact nneg_divr_closed : divr_closed (@nneg R).
#[export ]
HB.instance Definition _ := GRing.isDivClosed.Build R Rnneg_pred
  nneg_divr_closed.

Fact nneg_addr_closed : addr_closed (@nneg R).
#[export ]
HB.instance Definition _ := GRing.isAddClosed.Build R Rnneg_pred
  nneg_addr_closed.

Fact real_oppr_closed : oppr_closed (@real R).
#[export ]
HB.instance Definition _ := GRing.isOppClosed.Build R Rreal_pred
  real_oppr_closed.

Fact real_addr_closed : addr_closed (@real R).
#[export ]
HB.instance Definition _ := GRing.isAddClosed.Build R Rreal_pred
  real_addr_closed.

Fact real_divr_closed : divr_closed (@real R).
#[export ]
HB.instance Definition _ := GRing.isDivClosed.Build R Rreal_pred
  real_divr_closed.

End NumDomain.

Lemma num_real (R : realDomainType) (x : R) : x \is real.

Section RealClosed.
Variable R : rcfType.

Lemma poly_ivt : real_closed_axiom R.

Fact sqrtr_subproof (x : R) :
  exists2 y , 0 ≤ y & (if 0 ≤ x then y ^+ 2 == x else y == 0) : bool.

End RealClosed.

Module Exports. End Exports.

End Internals.

Module PredInstances.

Export Internals.Exports.

End PredInstances.

Module Import ExtraDef.

Definition sqrtr {R} x := s2val (sig2W (@sqrtr_subproof R x)).

End ExtraDef.

Notation sqrt := sqrtr.

Module Import Theory.

Section NumIntegralDomainTheory.

Variable R : numDomainType.
Implicit Types (V : semiNormedZmodType R) (x y z t : R).
Implicit Types (W : normedZmodType R).

Lemmas from the signature (reexported).

Definition ler_normD V (x y : V) : `|x + y | ≤ `|x | + `|y | :=
ler_normD x y.
Definition addr_gt0 x y : 0 < x → 0 < y → 0 < x + y := @addr_gt0 R x y.
Definition normr0_eq0 W (x : W) : `|x | = 0 → x = 0 := @normr0_eq0 R W x.
Definition ger_leVge x y : 0 ≤ x → 0 ≤ y → (x ≤ y ) || (y ≤ x ) :=
@ger_leVge R x y.
Definition normrM : {morph norm : x y / (x : R ) × y } := @normrM R.
Definition ler_def x y : (x ≤ y ) = (`|y - x | == y - x ) := ler_def x y.
Definition normrMn V (x : V) n : `|x *+ n | = `|x | *+ n := normrMn x n.
Definition normrN V (x : V) : `|- x | = `|x | := normrN x.

Predicate definitions.

Lemma posrE x : (x \is pos ) = (0 < x ).
Lemma negrE x : (x \is neg ) = (x < 0).
Lemma nnegrE x : (x \is nneg ) = (0 ≤ x ).
Lemma nposrE x : (x \is npos ) = (x ≤ 0).
Lemma realE x : (x \is real ) = (0 ≤ x ) || (x ≤ 0).

General properties of <= and <

Lemma lt0r x : (0 < x ) = (x != 0) && (0 ≤ x ).
Lemma le0r x : (0 ≤ x ) = (x == 0) || (0 < x ).

Lemma lt0r_neq0 (x : R) : 0 < x → x != 0.
Lemma ltr0_neq0 (x : R) : x < 0 → x != 0.

Lemma pmulr_rgt0 x y : 0 < x → (0 < x × y ) = (0 < y ).

Lemma pmulr_rge0 x y : 0 < x → (0 ≤ x × y ) = (0 ≤ y ).

Integer comparisons and characteristic 0.

Lemma ler01 : 0 ≤ 1 :> R.
Lemma ltr01 : 0 < 1 :> R.
Lemma ler0n n : 0 ≤ n %:R :> R.
Hint Extern 0 (is_true (@Order.le ring_display _ _ _)) ⇒
  (apply: ler01) : core.
Hint Extern 0 (is_true (@Order.lt ring_display _ _ _)) ⇒
  (apply: ltr01) : core.
Hint Extern 0 (is_true (@Order.le ring_display _ _ _)) ⇒
  (apply: ler0n) : core.
Lemma ltr0Sn n : 0 < n .+1 %:R :> R.
Lemma ltr0n n : (0 < n %:R :> R ) = (0 < n)%N.
Hint Extern 0 (is_true (@Order.lt ring_display _ _ _)) ⇒
  (apply: ltr0Sn) : core.

Lemma pnatr_eq0 n : (n %:R == 0 :> R ) = (n == 0)%N.

Lemma pchar_num : [pchar R ] =i pred0.

Properties of the norm.

Lemma ger0_def x : (0 ≤ x ) = (`|x | == x ).
Lemma normr_idP {x} : reflect (`|x | = x) (0 ≤ x).
Lemma ger0_norm x : 0 ≤ x → `|x | = x.
Lemma normr1 : `|1 : R | = 1.
Lemma normr_nat n : `|n %:R : R | = n %:R.

Lemma normr_prod I r (P : pred I) (F : I → R) :
  `|\prod_(i <- r | P i ) F i | = \prod_(i <- r | P i ) `|F i |.

Lemma normrX n x : `|x ^+ n | = `|x | ^+ n.

Lemma normr_unit : {homo (@norm _ (* R *) R ) : x / x \is a GRing.unit }.

Lemma normrV : {in GRing.unit , {morph (@norm _ (* R *) R ) : x / x ^-1 }}.

Lemma normrN1 : `|-1 : R | = 1.

Lemma big_real x0 op I (P : pred I) F (s : seq I) :
  {in real &, ∀ x y, op x y \is real } → x0 \is real →
  {in P , ∀ i, F i \is real } → \big [op /x0 ]_(i <- s | P i ) F i \is real.

Lemma sum_real I (P : pred I) (F : I → R) (s : seq I) :
  {in P , ∀ i, F i \is real } → \sum_(i <- s | P i ) F i \is real.

Lemma prod_real I (P : pred I) (F : I → R) (s : seq I) :
  {in P , ∀ i, F i \is real } → \prod_(i <- s | P i ) F i \is real.

Section SemiNormedZmoduleTheory.

Variable V : semiNormedZmodType R.
Implicit Types (v w : V).

Lemma normr0 : `|0 : V | = 0.

Lemma distrC v w : `|v - w | = `|w - v |.

Lemma normr_id v : `| `|v | | = `|v |.

Lemma normr_ge0 v : 0 ≤ `|v |.

Lemma normr_lt0 v : `|v | < 0 = false.

Lemma gtr0_norm_neq0 v : `|v | > 0 → (v != 0).

Lemma gtr0_norm_eq0F v : `|v | > 0 → (v == 0) = false.

End SemiNormedZmoduleTheory.

Section NormedZmoduleTheory.

Variable V : normedZmodType R.
Implicit Types (v w : V).

Lemma normr0P v : reflect (`|v | = 0) (v == 0).

Definition normr_eq0 v := sameP (`|v | =P 0) (normr0P v).

Lemma normr_le0 v : `|v | ≤ 0 = (v == 0).

Lemma normr_gt0 v : `|v | > 0 = (v != 0).

End NormedZmoduleTheory.

Definition normrE := (normr_id , normr0 , normr1 , normrN1 , normr_ge0 , normr_eq0 ,
  normr_lt0 , normr_le0 , normr_gt0 , normrN ).

Lemma ler0_def x : (x ≤ 0) = (`|x | == - x ).

Lemma ler0_norm x : x ≤ 0 → `|x | = - x.

Definition gtr0_norm x (hx : 0 < x) := ger0_norm (ltW hx).
Definition ltr0_norm x (hx : x < 0) := ler0_norm (ltW hx).

Lemma ger0_le_norm :
  {in nneg &, {mono (@normr _ R ) : x y / x ≤ y }}.

Lemma gtr0_le_norm :
  {in pos &, {mono (@normr _ R ) : x y / x ≤ y }}.

Lemma ler0_ge_norm :
  {in npos &, {mono (@normr _ R ) : x y / x ≤ y >-> x ≥ y }}.

Lemma ltr0_ge_norm :
  {in neg &, {mono (@normr _ R ) : x y / x ≤ y >-> x ≥ y }}.

Comparison to 0 of a difference

Lemma subr_ge0 x y : (0 ≤ y - x ) = (x ≤ y ).
Lemma subr_gt0 x y : (0 < y - x ) = (x < y ).
Lemma subr_le0 x y : (y - x ≤ 0) = (y ≤ x ).
(* FIXME: rewrite pattern *)
Lemma subr_lt0 x y : (y - x < 0) = (y < x ).
(* FIXME: rewrite pattern *)

Definition subr_lte0 := (subr_le0 , subr_lt0 ).
Definition subr_gte0 := (subr_ge0 , subr_gt0 ).
Definition subr_cp0 := (subr_lte0 , subr_gte0 ).

Comparability in a numDomain

Lemma comparable0r x : (0 >=< x)%R = (x \is Num.real ).

Lemma comparabler0 x : (x >=< 0)%R = (x \is Num.real ).

Lemma subr_comparable0 x y : (x - y >=< 0)%R = (x >=< y)%R.

Lemma comparablerE x y : (x >=< y)%R = (x - y \is Num.real ).

Lemma comparabler_trans : transitive (comparable : rel R).

Ordered ring properties.

Definition lter01 := (ler01 , ltr01 ).

Lemma addr_ge0 x y : 0 ≤ x → 0 ≤ y → 0 ≤ x + y.

End NumIntegralDomainTheory.

#[deprecated(since="mathcomp 2.4.0",note="Use pchar_num instead.")]
Notation char_num := pchar_num (only parsing).

Arguments ler01 {R}.
Arguments ltr01 {R}.
Arguments normr_idP {R x}.
Arguments normr0P {R V v}.
#[global] Hint Extern 0 (is_true (@Order.le ring_display _ _ _)) ⇒
  (apply: ler01) : core.
#[global] Hint Extern 0 (is_true (@Order.lt ring_display _ _ _)) ⇒
  (apply: ltr01) : core.
#[global] Hint Extern 0 (is_true (@Order.le ring_display _ _ _)) ⇒
  (apply: ler0n) : core.
#[global] Hint Extern 0 (is_true (@Order.lt ring_display _ _ _)) ⇒
  (apply: ltr0Sn) : core.
#[global] Hint Extern 0 (is_true (0 ≤ norm _)) ⇒ apply: normr_ge0 : core.

Lemma normr_nneg (R : numDomainType) (x : R) : `|x | \is Num.nneg.
#[global] Hint Resolve normr_nneg : core.

Section NumDomainOperationTheory.

Variable R : numDomainType.
Implicit Types x y z t : R.

Comparison and opposite.

Lemma lerN2 : {mono -%R : x y /~ x ≤ y :> R }.
Hint Resolve lerN2 : core.
Lemma ltrN2 : {mono -%R : x y /~ x < y :> R }.
Hint Resolve ltrN2 : core.
Definition lterN2 := (lerN2 , ltrN2 ).

Lemma lerNr x y : (x ≤ - y ) = (y ≤ - x ).

Lemma ltrNr x y : (x < - y ) = (y < - x ).

Definition lterNr := (lerNr , ltrNr ).

Lemma lerNl x y : (- x ≤ y ) = (- y ≤ x ).

Lemma ltrNl x y : (- x < y ) = (- y < x ).

Definition lterNl := (lerNl , ltrNl ).

Lemma oppr_ge0 x : (0 ≤ - x ) = (x ≤ 0).

Lemma oppr_gt0 x : (0 < - x ) = (x < 0).

Definition oppr_gte0 := (oppr_ge0 , oppr_gt0 ).

Lemma oppr_le0 x : (- x ≤ 0) = (0 ≤ x ).

Lemma oppr_lt0 x : (- x < 0) = (0 < x ).

Lemma gtrN x : 0 < x → - x < x.

Definition oppr_lte0 := (oppr_le0 , oppr_lt0 ).
Definition oppr_cp0 := (oppr_gte0 , oppr_lte0 ).
Definition lterNE := (oppr_cp0 , lterN2 ).

Lemma ge0_cp x : 0 ≤ x → (- x ≤ 0) × (- x ≤ x ).

Lemma gt0_cp x : 0 < x →
(0 ≤ x ) × (- x ≤ 0) × (- x ≤ x ) × (- x < 0) × (- x < x ).

Lemma le0_cp x : x ≤ 0 → (0 ≤ - x ) × (x ≤ - x ).

Lemma lt0_cp x :
x < 0 → (x ≤ 0) × (0 ≤ - x ) × (x ≤ - x ) × (0 < - x ) × (x < - x ).

Properties of the real subset.

Lemma ger0_real x : 0 ≤ x → x \is real.

Lemma ler0_real x : x ≤ 0 → x \is real.

Lemma gtr0_real x : 0 < x → x \is real.

Lemma ltr0_real x : x < 0 → x \is real.

Lemma real0 : 0 \is @real R.
Lemma real1 : 1 \is @real R.
Lemma realn n : n %:R \is @real R.
#[local] Hint Resolve real0 real1 : core.

Lemma ler_leVge x y : x ≤ 0 → y ≤ 0 → (x ≤ y ) || (y ≤ x ).

Lemma real_leVge x y : x \is real → y \is real → (x ≤ y ) || (y ≤ x ).

Lemma real_comparable x y : x \is real → y \is real → x >=< y.

Lemma realB : {in real &, ∀ x y, x - y \is real }.

Lemma realN : {mono (@GRing.opp R ) : x / x \is real }.

Lemma realBC x y : (x - y \is real ) = (y - x \is real ).

Lemma realD : {in real &, ∀ x y, x + y \is real }.

dichotomy and trichotomy

Variant ler_xor_gt (x y : R) : R → R → R → R → R → R →
    bool → bool → Set :=
  | LerNotGt of x ≤ y : ler_xor_gt x y x x y y (y - x) (y - x) true false
  | GtrNotLe of y < x : ler_xor_gt x y y y x x (x - y) (x - y) false true.

Variant ltr_xor_ge (x y : R) : R → R → R → R → R → R →
    bool → bool → Set :=
  | LtrNotGe of x < y : ltr_xor_ge x y x x y y (y - x) (y - x) false true
  | GerNotLt of y ≤ x : ltr_xor_ge x y y y x x (x - y) (x - y) true false.

Variant comparer x y : R → R → R → R → R → R →
    bool → bool → bool → bool → bool → bool → Set :=
  | ComparerLt of x < y : comparer x y x x y y (y - x) (y - x)
    false false false true false true
  | ComparerGt of x > y : comparer x y y y x x (x - y) (x - y)
    false false true false true false
  | ComparerEq of x = y : comparer x y x x x x 0 0
    true true true true false false.

Lemma real_leP x y : x \is real → y \is real →
  ler_xor_gt x y (min y x) (min x y) (max y x) (max x y)
                 `|x - y | `|y - x | (x ≤ y) (y < x).

Lemma real_ltP x y : x \is real → y \is real →
  ltr_xor_ge x y (min y x) (min x y) (max y x) (max x y)
             `|x - y | `|y - x | (y ≤ x) (x < y).

Lemma real_ltNge : {in real &, ∀ x y, (x < y ) = ~~ (y ≤ x )}.

Lemma real_leNgt : {in real &, ∀ x y, (x ≤ y ) = ~~ (y < x )}.

Lemma real_ltgtP x y : x \is real → y \is real →
  comparer x y (min y x) (min x y) (max y x) (max x y) `|x - y | `|y - x |
               (y == x) (x == y) (x ≥ y) (x ≤ y) (x > y) (x < y).

Variant ger0_xor_lt0 (x : R) : R → R → R → R → R →
    bool → bool → Set :=
  | Ger0NotLt0 of 0 ≤ x : ger0_xor_lt0 x 0 0 x x x false true
  | Ltr0NotGe0 of x < 0 : ger0_xor_lt0 x x x 0 0 (- x) true false.

Variant ler0_xor_gt0 (x : R) : R → R → R → R → R →
   bool → bool → Set :=
  | Ler0NotLe0 of x ≤ 0 : ler0_xor_gt0 x x x 0 0 (- x) false true
  | Gtr0NotGt0 of 0 < x : ler0_xor_gt0 x 0 0 x x x true false.

Variant comparer0 x : R → R → R → R → R →
    bool → bool → bool → bool → bool → bool → Set :=
  | ComparerGt0 of 0 < x : comparer0 x 0 0 x x x false false false true false true
  | ComparerLt0 of x < 0 : comparer0 x x x 0 0 (- x) false false true false true false
  | ComparerEq0 of x = 0 : comparer0 x 0 0 0 0 0 true true true true false false.

Lemma real_ge0P x : x \is real → ger0_xor_lt0 x
   (min 0 x) (min x 0) (max 0 x) (max x 0)
  `|x | (x < 0) (0 ≤ x).

Lemma real_le0P x : x \is real → ler0_xor_gt0 x
  (min 0 x) (min x 0) (max 0 x) (max x 0)
  `|x | (0 < x) (x ≤ 0).

Lemma real_ltgt0P x : x \is real →
  comparer0 x (min 0 x) (min x 0) (max 0 x) (max x 0)
            `|x | (0 == x) (x == 0) (x ≤ 0) (0 ≤ x) (x < 0) (x > 0).

Lemma max_real : {in real &, ∀ x y, max x y \is real }.

Lemma min_real : {in real &, ∀ x y, min x y \is real }.

Lemma bigmax_real I x0 (r : seq I) (P : pred I) (f : I → R):
  x0 \is real → {in P , ∀ i : I, f i \is real } →
  \big [max /x0 ]_(i <- r | P i ) f i \is real.

Lemma bigmin_real I x0 (r : seq I) (P : pred I) (f : I → R):
  x0 \is real → {in P , ∀ i : I, f i \is real } →
  \big [min /x0 ]_(i <- r | P i ) f i \is real.

Lemma real_neqr_lt : {in real &, ∀ x y, (x != y ) = (x < y ) || (y < x )}.

Lemma lerB_real x y : x ≤ y → y - x \is real.

Lemma gerB_real x y : x ≤ y → x - y \is real.

Lemma ler_real y x : x ≤ y → (x \is real ) = (y \is real ).

Lemma ger_real x y : y ≤ x → (x \is real ) = (y \is real ).

Lemma ger1_real x : 1 ≤ x → x \is real.
Lemma ler1_real x : x ≤ 1 → x \is real.

Lemma Nreal_leF x y : y \is real → x \notin real → (x ≤ y ) = false.

Lemma Nreal_geF x y : y \is real → x \notin real → (y ≤ x ) = false.

Lemma Nreal_ltF x y : y \is real → x \notin real → (x < y ) = false.

Lemma Nreal_gtF x y : y \is real → x \notin real → (y < x ) = false.

real wlog

Lemma real_wlog_ler P :
    (∀ a b, P b a → P a b ) → (∀ a b, a ≤ b → P a b ) →
  ∀ a b : R, a \is real → b \is real → P a b.

Lemma real_wlog_ltr P :
    (∀ a, P a a ) → (∀ a b, (P b a → P a b)) →
    (∀ a b, a < b → P a b ) →
  ∀ a b : R, a \is real → b \is real → P a b.

Monotony of addition

Lemma lerD2l x : {mono +%R x : y z / y ≤ z }.

Lemma lerD2r x : {mono +%R ^~ x : y z / y ≤ z }.

Lemma ltrD2l x : {mono +%R x : y z / y < z }.

Lemma ltrD2r x : {mono +%R ^~ x : y z / y < z }.

Definition lerD2 := (lerD2l , lerD2r ).
Definition ltrD2 := (ltrD2l , ltrD2r ).
Definition lterD2 := (lerD2 , ltrD2 ).

Addition, subtraction and transitivity

Lemma lerD x y z t : x ≤ y → z ≤ t → x + z ≤ y + t.

Lemma ler_ltD x y z t : x ≤ y → z < t → x + z < y + t.

Lemma ltr_leD x y z t : x < y → z ≤ t → x + z < y + t.

Lemma ltrD x y z t : x < y → z < t → x + z < y + t.

Lemma lerB x y z t : x ≤ y → t ≤ z → x - z ≤ y - t.

Lemma ler_ltB x y z t : x ≤ y → t < z → x - z < y - t.

Lemma ltr_leB x y z t : x < y → t ≤ z → x - z < y - t.

Lemma ltrB x y z t : x < y → t < z → x - z < y - t.

Lemma lerBlDr x y z : (x - y ≤ z ) = (x ≤ z + y ).

Lemma ltrBlDr x y z : (x - y < z ) = (x < z + y ).

Lemma lerBrDr x y z : (x ≤ y - z ) = (x + z ≤ y ).

Lemma ltrBrDr x y z : (x < y - z ) = (x + z < y ).

Definition lerBDr := (lerBlDr , lerBrDr ).
Definition ltrBDr := (ltrBlDr , ltrBrDr ).
Definition lterBDr := (lerBDr , ltrBDr ).

Lemma lerBlDl x y z : (x - y ≤ z ) = (x ≤ y + z ).

Lemma ltrBlDl x y z : (x - y < z ) = (x < y + z ).

Lemma lerBrDl x y z : (x ≤ y - z ) = (z + x ≤ y ).

Lemma ltrBrDl x y z : (x < y - z ) = (z + x < y ).

Definition lerBDl := (lerBlDl , lerBrDl ).
Definition ltrBDl := (ltrBlDl , ltrBrDl ).
Definition lterBDl := (lerBDl , ltrBDl ).

Lemma lerDl x y : (x ≤ x + y ) = (0 ≤ y ).

Lemma ltrDl x y : (x < x + y ) = (0 < y ).

Lemma lerDr x y : (x ≤ y + x ) = (0 ≤ y ).

Lemma ltrDr x y : (x < y + x ) = (0 < y ).

Lemma gerDl x y : (x + y ≤ x ) = (y ≤ 0).

Lemma gerBl x y : (x - y ≤ x ) = (0 ≤ y ).

Lemma gtrDl x y : (x + y < x ) = (y < 0).

Lemma gtrBl x y : (x - y < x ) = (0 < y ).

Lemma gerDr x y : (y + x ≤ x ) = (y ≤ 0).

Lemma gtrDr x y : (y + x < x ) = (y < 0).

Definition cprD := (lerDl , lerDr , gerDl , gerDl ,
ltrDl , ltrDr , gtrDl , gtrDl ).

Addition with left member known to be positive/negative

Lemma ler_wpDl y x z : 0 ≤ x → y ≤ z → y ≤ x + z.

Lemma ltr_wpDl y x z : 0 ≤ x → y < z → y < x + z.

Lemma ltr_pwDl y x z : 0 < x → y ≤ z → y < x + z.

Lemma ltr_pDl y x z : 0 < x → y < z → y < x + z.

Lemma ler_wnDl y x z : x ≤ 0 → y ≤ z → x + y ≤ z.

Lemma ltr_wnDl y x z : x ≤ 0 → y < z → x + y < z.

Lemma ltr_nwDl y x z : x < 0 → y ≤ z → x + y < z.

Lemma ltr_nDl y x z : x < 0 → y < z → x + y < z.

Addition with right member we know positive/negative

Lemma ler_wpDr y x z : 0 ≤ x → y ≤ z → y ≤ z + x.

Lemma ltr_wpDr y x z : 0 ≤ x → y < z → y < z + x.

Lemma ltr_pwDr y x z : 0 < x → y ≤ z → y < z + x.

Lemma ltr_pDr y x z : 0 < x → y < z → y < z + x.

Lemma ler_wnDr y x z : x ≤ 0 → y ≤ z → y + x ≤ z.

Lemma ltr_wnDr y x z : x ≤ 0 → y < z → y + x < z.

Lemma ltr_nwDr y x z : x < 0 → y ≤ z → y + x < z.

Lemma ltr_nDr y x z : x < 0 → y < z → y + x < z.

x and y have the same sign and their sum is null

Lemma paddr_eq0 (x y : R) :
  0 ≤ x → 0 ≤ y → (x + y == 0) = (x == 0) && (y == 0).

Lemma naddr_eq0 (x y : R) :
  x ≤ 0 → y ≤ 0 → (x + y == 0) = (x == 0) && (y == 0).

Lemma addr_ss_eq0 (x y : R) :
    (0 ≤ x ) && (0 ≤ y ) || (x ≤ 0) && (y ≤ 0) →
  (x + y == 0) = (x == 0) && (y == 0).

big sum and ler

Lemma sumr_ge0 I (r : seq I) (P : pred I) (F : I → R) :
  (∀ i, P i → (0 ≤ F i )) → 0 ≤ \sum_(i <- r | P i ) (F i ).

Lemma ler_sum I (r : seq I) (P : pred I) (F G : I → R) :
    (∀ i, P i → F i ≤ G i ) →
  \sum_(i <- r | P i ) F i ≤ \sum_(i <- r | P i ) G i.

Lemma ler_sum_nat (m n : nat) (F G : nat → R) :
  (∀ i, (m ≤ i < n)%N → F i ≤ G i ) →
  \sum_(m ≤ i < n ) F i ≤ \sum_(m ≤ i < n ) G i.

Lemma psumr_eq0 (I : eqType) (r : seq I) (P : pred I) (F : I → R) :
    (∀ i, P i → 0 ≤ F i ) →
  (\sum_(i <- r | P i ) (F i ) == 0) = (all (fun i ⇒ (P i ) ==> (F i == 0)) r ).

:TODO: Cyril : See which form to keep

Lemma psumr_eq0P (I : finType) (P : pred I) (F : I → R) :
     (∀ i, P i → 0 ≤ F i ) → \sum_(i | P i ) F i = 0 →
  (∀ i, P i → F i = 0).

Lemma psumr_neq0 (I : eqType) (r : seq I) (P : pred I) (F : I → R) :
    (∀ i, P i → 0 ≤ F i ) →
  (\sum_(i <- r | P i ) (F i ) != 0) = (has (fun i ⇒ P i && (0 < F i )) r ).

Lemma psumr_neq0P (I : finType) (P : pred I) (F : I → R) :
     (∀ i, P i → 0 ≤ F i ) → \sum_(i | P i ) F i ≠ 0 →
  (∃ i , P i && (0 < F i )).

mulr and ler/ltr

Lemma ler_pM2l x : 0 < x → {mono *%R x : x y / x ≤ y }.

Lemma ltr_pM2l x : 0 < x → {mono *%R x : x y / x < y }.

Definition lter_pM2l := (ler_pM2l , ltr_pM2l ).

Lemma ler_pM2r x : 0 < x → {mono *%R ^~ x : x y / x ≤ y }.

Lemma ltr_pM2r x : 0 < x → {mono *%R ^~ x : x y / x < y }.

Definition lter_pM2r := (ler_pM2r , ltr_pM2r ).

Lemma ler_nM2l x : x < 0 → {mono *%R x : x y /~ x ≤ y }.

Lemma ltr_nM2l x : x < 0 → {mono *%R x : x y /~ x < y }.

Definition lter_nM2l := (ler_nM2l , ltr_nM2l ).

Lemma ler_nM2r x : x < 0 → {mono *%R ^~ x : x y /~ x ≤ y }.

Lemma ltr_nM2r x : x < 0 → {mono *%R ^~ x : x y /~ x < y }.

Definition lter_nM2r := (ler_nM2r , ltr_nM2r ).

Lemma ler_wpM2l x : 0 ≤ x → {homo *%R x : y z / y ≤ z }.

Lemma ler_wpM2r x : 0 ≤ x → {homo *%R ^~ x : y z / y ≤ z }.

Lemma ler_wnM2l x : x ≤ 0 → {homo *%R x : y z /~ y ≤ z }.

Lemma ler_wnM2r x : x ≤ 0 → {homo *%R ^~ x : y z /~ y ≤ z }.

Binary forms, for backchaining.

Lemma ler_pM x1 y1 x2 y2 :
0 ≤ x1 → 0 ≤ x2 → x1 ≤ y1 → x2 ≤ y2 → x1 × x2 ≤ y1 × y2.

Lemma ltr_pM x1 y1 x2 y2 :
0 ≤ x1 → 0 ≤ x2 → x1 < y1 → x2 < y2 → x1 × x2 < y1 × y2.

complement for x *+ n and <= or <

Lemma ler_pMn2r n : (0 < n)%N → {mono (@GRing.natmul R )^~ n : x y / x ≤ y }.

Lemma ltr_pMn2r n : (0 < n)%N → {mono (@GRing.natmul R )^~ n : x y / x < y }.

Lemma pmulrnI n : (0 < n)%N → injective ((@GRing.natmul R )^~ n).

Lemma eqr_pMn2r n : (0 < n)%N → {mono (@GRing.natmul R )^~ n : x y / x == y }.

Lemma pmulrn_lgt0 x n : (0 < n)%N → (0 < x *+ n ) = (0 < x ).

Lemma pmulrn_llt0 x n : (0 < n)%N → (x *+ n < 0) = (x < 0).

Lemma pmulrn_lge0 x n : (0 < n)%N → (0 ≤ x *+ n ) = (0 ≤ x ).

Lemma pmulrn_lle0 x n : (0 < n)%N → (x *+ n ≤ 0) = (x ≤ 0).

Lemma ltr_wMn2r x y n : x < y → (x *+ n < y *+ n ) = (0 < n)%N.

Lemma ltr_wpMn2r n : (0 < n)%N → {homo (@GRing.natmul R )^~ n : x y / x < y }.

Lemma ler_wMn2r n : {homo (@GRing.natmul R )^~ n : x y / x ≤ y }.

Lemma mulrn_wge0 x n : 0 ≤ x → 0 ≤ x *+ n.

Lemma mulrn_wle0 x n : x ≤ 0 → x *+ n ≤ 0.

Lemma lerMn2r n x y : (x *+ n ≤ y *+ n ) = ((n == 0) || (x ≤ y )).

Lemma ltrMn2r n x y : (x *+ n < y *+ n ) = ((0 < n)%N && (x < y )).

Lemma eqrMn2r n x y : (x *+ n == y *+ n ) = (n == 0)%N || (x == y ).

More characteristic zero properties.

Lemma mulrn_eq0 x n : (x *+ n == 0) = ((n == 0)%N || (x == 0)).

Lemma eqNr x : (- x == x ) = (x == 0).

Lemma mulrIn x : x != 0 → injective (GRing.natmul x).

Lemma ler_wpMn2l x :
  0 ≤ x → {homo (@GRing.natmul R x ) : m n / (m ≤ n)%N >-> m ≤ n }.

Lemma ler_wnMn2l x :
  x ≤ 0 → {homo (@GRing.natmul R x ) : m n / (n ≤ m)%N >-> m ≤ n }.

Lemma mulrn_wgt0 x n : 0 < x → 0 < x *+ n = (0 < n)%N.

Lemma mulrn_wlt0 x n : x < 0 → x *+ n < 0 = (0 < n)%N.

Lemma ler_pMn2l x :
  0 < x → {mono (@GRing.natmul R x ) : m n / (m ≤ n)%N >-> m ≤ n }.

Lemma ltr_pMn2l x :
  0 < x → {mono (@GRing.natmul R x ) : m n / (m < n)%N >-> m < n }.

Lemma ler_nMn2l x :
  x < 0 → {mono (@GRing.natmul R x ) : m n / (n ≤ m)%N >-> m ≤ n }.

Lemma ltr_nMn2l x :
  x < 0 → {mono (@GRing.natmul R x ) : m n / (n < m)%N >-> m < n }.

Lemma ler_nat m n : (m %:R ≤ n %:R :> R ) = (m ≤ n)%N.

Lemma ltr_nat m n : (m %:R < n %:R :> R ) = (m < n)%N.

Lemma eqr_nat m n : (m %:R == n %:R :> R ) = (m == n)%N.

Lemma pnatr_eq1 n : (n %:R == 1 :> R ) = (n == 1)%N.

Lemma lern0 n : (n %:R ≤ 0 :> R ) = (n == 0).

Lemma ltrn0 n : (n %:R < 0 :> R ) = false.

Lemma ler1n n : 1 ≤ n %:R :> R = (1 ≤ n)%N.
Lemma ltr1n n : 1 < n %:R :> R = (1 < n)%N.
Lemma lern1 n : n %:R ≤ 1 :> R = (n ≤ 1)%N.
Lemma ltrn1 n : n %:R < 1 :> R = (n < 1)%N.

Lemma ltrN10 : -1 < 0 :> R.
Lemma lerN10 : -1 ≤ 0 :> R.
Lemma ltr10 : 1 < 0 :> R = false.
Lemma ler10 : 1 ≤ 0 :> R = false.
Lemma ltr0N1 : 0 < -1 :> R = false.
Lemma ler0N1 : 0 ≤ -1 :> R = false.

#[deprecated(since="mathcomp 2.4.0", note="use `mulrn_wgt0` instead")]
Lemma pmulrn_rgt0 x n : 0 < x → 0 < x *+ n = (0 < n)%N.

Lemma pmulrn_rlt0 x n : 0 < x → x *+ n < 0 = false.

Lemma pmulrn_rge0 x n : 0 < x → 0 ≤ x *+ n.

Lemma pmulrn_rle0 x n : 0 < x → x *+ n ≤ 0 = (n == 0)%N.

Lemma nmulrn_rgt0 x n : x < 0 → 0 < x *+ n = false.

Lemma nmulrn_rge0 x n : x < 0 → 0 ≤ x *+ n = (n == 0)%N.

Lemma nmulrn_rle0 x n : x < 0 → x *+ n ≤ 0.

(x * y) compared to 0 Remark : pmulr_rgt0 and pmulr_rge0 are defined above x positive and y right

Lemma pmulr_rlt0 x y : 0 < x → (x × y < 0) = (y < 0).

Lemma pmulr_rle0 x y : 0 < x → (x × y ≤ 0) = (y ≤ 0).

x positive and y left

Lemma pmulr_lgt0 x y : 0 < x → (0 < y × x ) = (0 < y ).

Lemma pmulr_lge0 x y : 0 < x → (0 ≤ y × x ) = (0 ≤ y ).

Lemma pmulr_llt0 x y : 0 < x → (y × x < 0) = (y < 0).

Lemma pmulr_lle0 x y : 0 < x → (y × x ≤ 0) = (y ≤ 0).

x negative and y right

Lemma nmulr_rgt0 x y : x < 0 → (0 < x × y ) = (y < 0).

Lemma nmulr_rge0 x y : x < 0 → (0 ≤ x × y ) = (y ≤ 0).

Lemma nmulr_rlt0 x y : x < 0 → (x × y < 0) = (0 < y ).

Lemma nmulr_rle0 x y : x < 0 → (x × y ≤ 0) = (0 ≤ y ).

x negative and y left

Lemma nmulr_lgt0 x y : x < 0 → (0 < y × x ) = (y < 0).

Lemma nmulr_lge0 x y : x < 0 → (0 ≤ y × x ) = (y ≤ 0).

Lemma nmulr_llt0 x y : x < 0 → (y × x < 0) = (0 < y ).

Lemma nmulr_lle0 x y : x < 0 → (y × x ≤ 0) = (0 ≤ y ).

weak and symmetric lemmas

Lemma mulr_ge0 x y : 0 ≤ x → 0 ≤ y → 0 ≤ x × y.

Lemma mulr_le0 x y : x ≤ 0 → y ≤ 0 → 0 ≤ x × y.

Lemma mulr_ge0_le0 x y : 0 ≤ x → y ≤ 0 → x × y ≤ 0.

Lemma mulr_le0_ge0 x y : x ≤ 0 → 0 ≤ y → x × y ≤ 0.

mulr_gt0 with only one case

Lemma mulr_gt0 x y : 0 < x → 0 < y → 0 < x × y.

and reverse direction

Lemma mulr_ge0_gt0 x y : 0 ≤ x → 0 ≤ y → (0 < x × y ) = (0 < x ) && (0 < y ).

Iterated products

Lemma prodr_ge0 I r (P : pred I) (E : I → R) :
  (∀ i, P i → 0 ≤ E i ) → 0 ≤ \prod_(i <- r | P i ) E i.

Lemma prodr_gt0 I r (P : pred I) (E : I → R) :
  (∀ i, P i → 0 < E i ) → 0 < \prod_(i <- r | P i ) E i.

Lemma ler_prod I r (P : pred I) (E1 E2 : I → R) :
    (∀ i, P i → 0 ≤ E1 i ≤ E2 i ) →
  \prod_(i <- r | P i ) E1 i ≤ \prod_(i <- r | P i ) E2 i.

Lemma ltr_prod I r (P : pred I) (E1 E2 : I → R) :
    has P r → (∀ i, P i → 0 ≤ E1 i < E2 i ) →
  \prod_(i <- r | P i ) E1 i < \prod_(i <- r | P i ) E2 i.

Lemma ltr_prod_nat (E1 E2 : nat → R) (n m : nat) :
   (m < n)%N → (∀ i, (m ≤ i < n)%N → 0 ≤ E1 i < E2 i ) →
  \prod_(m ≤ i < n ) E1 i < \prod_(m ≤ i < n ) E2 i.

real of mul

Lemma realMr x y : x != 0 → x \is real → (x × y \is real ) = (y \is real ).

Lemma realrM x y : y != 0 → y \is real → (x × y \is real ) = (x \is real ).

Lemma realM : {in real &, ∀ x y, x × y \is real }.

Lemma realrMn x n : (n != 0)%N → (x *+ n \is real ) = (x \is real ).

ler/ltr and multiplication between a positive/negative

Lemma ger_pMl x y : 0 < y → (x × y ≤ y ) = (x ≤ 1).

Lemma gtr_pMl x y : 0 < y → (x × y < y ) = (x < 1).

Lemma ger_pMr x y : 0 < y → (y × x ≤ y ) = (x ≤ 1).

Lemma gtr_pMr x y : 0 < y → (y × x < y ) = (x < 1).

Lemma ler_pMl x y : 0 < y → (y ≤ x × y ) = (1 ≤ x ).

Lemma ltr_pMl x y : 0 < y → (y < x × y ) = (1 < x ).

Lemma ler_pMr x y : 0 < y → (y ≤ y × x ) = (1 ≤ x ).

Lemma ltr_pMr x y : 0 < y → (y < y × x ) = (1 < x ).

Lemma ger_nMl x y : y < 0 → (x × y ≤ y ) = (1 ≤ x ).

Lemma gtr_nMl x y : y < 0 → (x × y < y ) = (1 < x ).

Lemma ger_nMr x y : y < 0 → (y × x ≤ y ) = (1 ≤ x ).

Lemma gtr_nMr x y : y < 0 → (y × x < y ) = (1 < x ).

Lemma ler_nMl x y : y < 0 → (y ≤ x × y ) = (x ≤ 1).

Lemma ltr_nMl x y : y < 0 → (y < x × y ) = (x < 1).

Lemma ler_nMr x y : y < 0 → (y ≤ y × x ) = (x ≤ 1).

Lemma ltr_nMr x y : y < 0 → (y < y × x ) = (x < 1).

ler/ltr and multiplication between a positive/negative and a exterior (1 <= _) or interior (0 <= _ <= 1)

Lemma ler_peMl x y : 0 ≤ y → 1 ≤ x → y ≤ x × y.

Lemma ler_neMl x y : y ≤ 0 → 1 ≤ x → x × y ≤ y.

Lemma ler_peMr x y : 0 ≤ y → 1 ≤ x → y ≤ y × x.

Lemma ler_neMr x y : y ≤ 0 → 1 ≤ x → y × x ≤ y.

Lemma ler_piMl x y : 0 ≤ y → x ≤ 1 → x × y ≤ y.

Lemma ler_niMl x y : y ≤ 0 → x ≤ 1 → y ≤ x × y.

Lemma ler_piMr x y : 0 ≤ y → x ≤ 1 → y × x ≤ y.

Lemma ler_niMr x y : y ≤ 0 → x ≤ 1 → y ≤ y × x.

Lemma mulr_ile1 x y : 0 ≤ x → 0 ≤ y → x ≤ 1 → y ≤ 1 → x × y ≤ 1.

Lemma mulr_ilt1 x y : 0 ≤ x → 0 ≤ y → x < 1 → y < 1 → x × y < 1.

Definition mulr_ilte1 := (mulr_ile1 , mulr_ilt1 ).

Lemma mulr_ege1 x y : 1 ≤ x → 1 ≤ y → 1 ≤ x × y.

Lemma mulr_egt1 x y : 1 < x → 1 < y → 1 < x × y.
Definition mulr_egte1 := (mulr_ege1 , mulr_egt1 ).
Definition mulr_cp1 := (mulr_ilte1 , mulr_egte1 ).

ler and ^-1

Lemma invr_gt0 x : (0 < x ^-1 ) = (0 < x ).

Lemma invr_ge0 x : (0 ≤ x ^-1 ) = (0 ≤ x ).

Lemma invr_lt0 x : (x ^-1 < 0) = (x < 0).

Lemma invr_le0 x : (x ^-1 ≤ 0) = (x ≤ 0).

Definition invr_gte0 := (invr_ge0 , invr_gt0 ).
Definition invr_lte0 := (invr_le0 , invr_lt0 ).

Lemma divr_ge0 x y : 0 ≤ x → 0 ≤ y → 0 ≤ x / y.

Lemma divr_gt0 x y : 0 < x → 0 < y → 0 < x / y.

Lemma realV : {mono (@GRing.inv R ) : x / x \is real }.

ler and exprn

Lemma exprn_ge0 n x : 0 ≤ x → 0 ≤ x ^+ n.

Lemma realX n : {in real , ∀ x, x ^+ n \is real }.

Lemma exprn_gt0 n x : 0 < x → 0 < x ^+ n.

Definition exprn_gte0 := (exprn_ge0 , exprn_gt0 ).

Lemma exprn_ile1 n x : 0 ≤ x → x ≤ 1 → x ^+ n ≤ 1.

Lemma exprn_ilt1 n x : 0 ≤ x → x < 1 → x ^+ n < 1 = (n != 0).

Definition exprn_ilte1 := (exprn_ile1 , exprn_ilt1 ).

Lemma exprn_ege1 n x : 1 ≤ x → 1 ≤ x ^+ n.

Lemma exprn_egt1 n x : 1 < x → 1 < x ^+ n = (n != 0).

Definition exprn_egte1 := (exprn_ege1 , exprn_egt1 ).
Definition exprn_cp1 := (exprn_ilte1 , exprn_egte1 ).

Lemma ler_iXnr x n : (0 < n)%N → 0 ≤ x → x ≤ 1 → x ^+ n ≤ x.

Lemma ltr_iXnr x n : 0 < x → x < 1 → (x ^+ n < x ) = (1 < n)%N.

Definition lter_iXnr := (ler_iXnr , ltr_iXnr ).

Lemma ler_eXnr x n : (0 < n)%N → 1 ≤ x → x ≤ x ^+ n.

Lemma ltr_eXnr x n : 1 < x → (x < x ^+ n ) = (1 < n)%N.

Definition lter_eXnr := (ler_eXnr , ltr_eXnr ).
Definition lter_Xnr := (lter_iXnr , lter_eXnr ).

Lemma ler_wiXn2l x :
  0 ≤ x → x ≤ 1 → {homo GRing.exp x : m n / (n ≤ m)%N >-> m ≤ n }.

Lemma ler_weXn2l x : 1 ≤ x → {homo GRing.exp x : m n / (m ≤ n)%N >-> m ≤ n }.

Lemma ieexprn_weq1 x n : 0 ≤ x → (x ^+ n == 1) = ((n == 0) || (x == 1)).

Lemma ieexprIn x : 0 < x → x != 1 → injective (GRing.exp x).

Lemma ler_iXn2l x :
  0 < x → x < 1 → {mono GRing.exp x : m n / (n ≤ m)%N >-> m ≤ n }.

Lemma ltr_iXn2l x :
  0 < x → x < 1 → {mono GRing.exp x : m n / (n < m)%N >-> m < n }.

Definition lter_iXn2l := (ler_iXn2l , ltr_iXn2l ).

Lemma ler_eXn2l x :
  1 < x → {mono GRing.exp x : m n / (m ≤ n)%N >-> m ≤ n }.

Lemma ltr_eXn2l x :
  1 < x → {mono (GRing.exp x ) : m n / (m < n)%N >-> m < n }.

Definition lter_eXn2l := (ler_eXn2l , ltr_eXn2l ).

Lemma ltrXn2r n x y : 0 ≤ x → x < y → x ^+ n < y ^+ n = (n != 0).

Lemma lerXn2r n : {in nneg & , {homo (@GRing.exp R )^~ n : x y / x ≤ y }}.

Definition lterXn2r := (lerXn2r , ltrXn2r ).

Lemma ltr_wpXn2r n :
  (0 < n)%N → {in nneg & , {homo (@GRing.exp R )^~ n : x y / x < y }}.

Lemma ler_pXn2r n :
  (0 < n)%N → {in nneg & , {mono (@GRing.exp R )^~ n : x y / x ≤ y }}.

Lemma ltr_pXn2r n :
  (0 < n)%N → {in nneg & , {mono (@GRing.exp R )^~ n : x y / x < y }}.

Definition lter_pXn2r := (ler_pXn2r , ltr_pXn2r ).

Lemma pexpIrn n : (0 < n)%N → {in nneg &, injective ((@GRing.exp R )^~ n)}.

expr and ler/ltr

Lemma expr_le1 n x : (0 < n)%N → 0 ≤ x → (x ^+ n ≤ 1) = (x ≤ 1).

Lemma expr_lt1 n x : (0 < n)%N → 0 ≤ x → (x ^+ n < 1) = (x < 1).

Definition expr_lte1 := (expr_le1 , expr_lt1 ).

Lemma expr_ge1 n x : (0 < n)%N → 0 ≤ x → (1 ≤ x ^+ n ) = (1 ≤ x ).

Lemma expr_gt1 n x : (0 < n)%N → 0 ≤ x → (1 < x ^+ n ) = (1 < x ).

Definition expr_gte1 := (expr_ge1 , expr_gt1 ).

Lemma pexpr_eq1 x n : (0 < n)%N → 0 ≤ x → (x ^+ n == 1) = (x == 1).

Lemma pexprn_eq1 x n : 0 ≤ x → (x ^+ n == 1) = (n == 0) || (x == 1).

Lemma eqrXn2 n x y :
  (0 < n)%N → 0 ≤ x → 0 ≤ y → (x ^+ n == y ^+ n ) = (x == y ).

Lemma sqrp_eq1 x : 0 ≤ x → (x ^+ 2 == 1) = (x == 1).

Lemma sqrn_eq1 x : x ≤ 0 → (x ^+ 2 == 1) = (x == -1).

Lemma ler_sqr : {in nneg &, {mono (fun x ⇒ x ^+ 2) : x y / x ≤ y }}.

Lemma ltr_sqr : {in nneg &, {mono (fun x ⇒ x ^+ 2) : x y / x < y }}.

Lemma ler_pV2 :
  {in [pred x in GRing.unit | 0 < x ] &, {mono (@GRing.inv R ) : x y /~ x ≤ y }}.

Lemma ler_nV2 :
  {in [pred x in GRing.unit | x < 0] &, {mono (@GRing.inv R ) : x y /~ x ≤ y }}.

Lemma ltr_pV2 :
  {in [pred x in GRing.unit | 0 < x ] &, {mono (@GRing.inv R ) : x y /~ x < y }}.

Lemma ltr_nV2 :
  {in [pred x in GRing.unit | x < 0] &, {mono (@GRing.inv R ) : x y /~ x < y }}.

Lemma invr_gt1 x : x \is a GRing.unit → 0 < x → (1 < x ^-1 ) = (x < 1).

Lemma invr_ge1 x : x \is a GRing.unit → 0 < x → (1 ≤ x ^-1 ) = (x ≤ 1).

Definition invr_gte1 := (invr_ge1 , invr_gt1 ).

Lemma invr_le1 x (ux : x \is a GRing.unit) (hx : 0 < x) :
  (x ^-1 ≤ 1) = (1 ≤ x ).

Lemma invr_lt1 x (ux : x \is a GRing.unit) (hx : 0 < x) : (x ^-1 < 1) = (1 < x ).

Definition invr_lte1 := (invr_le1 , invr_lt1 ).
Definition invr_cp1 := (invr_gte1 , invr_lte1 ).

max and min

Lemma natr_min (m n : nat) : (Order.min m n )%:R = Order.min m %:R n %:R :> R.

Lemma natr_max (m n : nat) : (Order.max m n )%:R = Order.max m %:R n %:R :> R.

Lemma addr_min_max x y : min x y + max x y = x + y.

Lemma addr_max_min x y : max x y + min x y = x + y.

Lemma minr_to_max x y : min x y = x + y - max x y.

Lemma maxr_to_min x y : max x y = x + y - min x y.

Lemma real_oppr_max : {in real &, {morph -%R : x y / max x y >-> min x y : R }}.

Lemma real_oppr_min : {in real &, {morph -%R : x y / min x y >-> max x y : R }}.

Lemma real_addr_minl : {in real & real & real , @left_distributive R R +%R min }.

Lemma real_addr_minr : {in real & real & real , @right_distributive R R +%R min }.

Lemma real_addr_maxl : {in real & real & real , @left_distributive R R +%R max }.

Lemma real_addr_maxr : {in real & real & real , @right_distributive R R +%R max }.

Lemma minr_pMr x y z : 0 ≤ x → x × min y z = min (x × y) (x × z).

Lemma maxr_pMr x y z : 0 ≤ x → x × max y z = max (x × y) (x × z).

Lemma real_maxr_nMr x y z : x ≤ 0 → y \is real → z \is real →
  x × max y z = min (x × y) (x × z).

Lemma real_minr_nMr x y z : x ≤ 0 → y \is real → z \is real →
  x × min y z = max (x × y) (x × z).

Lemma minr_pMl x y z : 0 ≤ x → min y z × x = min (y × x) (z × x).

Lemma maxr_pMl x y z : 0 ≤ x → max y z × x = max (y × x) (z × x).

Lemma real_minr_nMl x y z : x ≤ 0 → y \is real → z \is real →
  min y z × x = max (y × x) (z × x).

Lemma real_maxr_nMl x y z : x ≤ 0 → y \is real → z \is real →
  max y z × x = min (y × x) (z × x).

Lemma real_maxrN x : x \is real → max x (- x) = `|x |.

Lemma real_maxNr x : x \is real → max (- x) x = `|x |.

Lemma real_minrN x : x \is real → min x (- x) = - `|x |.

Lemma real_minNr x : x \is real → min (- x) x = - `|x |.

Section RealDomainArgExtremum.

Context {I : finType} (i0 : I).
Context (P : pred I) (F : I → R) (Pi0 : P i0).
Hypothesis F_real : {in P , ∀ i, F i \is real }.

Lemma real_arg_minP : extremum_spec <=%R P F [arg min_(i < i0 | P i ) F i ].

Lemma real_arg_maxP : extremum_spec >=%R P F [arg max_(i > i0 | P i ) F i ].

End RealDomainArgExtremum.

norm

Lemma real_ler_norm x : x \is real → x ≤ `|x |.

norm + add

GG: Could this be a better definition of "real" ?

Lemma realEsqr x : (x \is real ) = (0 ≤ x ^+ 2).

Lemma real_normK x : x \is real → `|x | ^+ 2 = x ^+ 2.

Binary sign ((-1) ^+ s).

Lemma normr_sign s : `|(-1) ^+ s : R | = 1.

Lemma normrMsign s x : `|(-1) ^+ s × x | = `|x |.

Lemma signr_gt0 (b : bool) : (0 < (-1) ^+ b :> R ) = ~~ b.

Lemma signr_lt0 (b : bool) : ((-1) ^+ b < 0 :> R ) = b.

Lemma signr_ge0 (b : bool) : (0 ≤ (-1) ^+ b :> R ) = ~~ b.

Lemma signr_le0 (b : bool) : ((-1) ^+ b ≤ 0 :> R ) = b.

This actually holds for char R != 2.

Lemma signr_inj : injective (fun b : bool ⇒ (-1) ^+ b : R).

Ternary sign (sg).

Lemma sgr_def x : sg x = (-1) ^+ (x < 0)%R *+ (x != 0).

Lemma neqr0_sign x : x != 0 → (-1) ^+ (x < 0)%R = sgr x.

Lemma gtr0_sg x : 0 < x → sg x = 1.

Lemma ltr0_sg x : x < 0 → sg x = -1.

Lemma sgr0 : sg 0 = 0 :> R.
Lemma sgr1 : sg 1 = 1 :> R.
Lemma sgrN1 : sg (-1) = -1 :> R.
Definition sgrE := (sgr0 , sgr1 , sgrN1 ).

Lemma sqr_sg x : sg x ^+ 2 = (x != 0)%:R.

Lemma mulr_sg_eq1 x y : (sg x × y == 1) = (x != 0) && (sg x == y ).

Lemma mulr_sg_eqN1 x y : (sg x × sg y == -1) = (x != 0) && (sg x == - sg y ).

Lemma sgr_eq0 x : (sg x == 0) = (x == 0).

Lemma sgr_odd n x : x != 0 → (sg x ) ^+ n = (sg x ) ^+ (odd n ).

Lemma sgrMn x n : sg (x *+ n) = (n != 0)%:R × sg x.

Lemma sgr_nat n : sg n %:R = (n != 0)%:R :> R.

Lemma sgr_id x : sg (sg x) = sg x.

Lemma sgr_lt0 x : (sg x < 0) = (x < 0).

Lemma sgr_le0 x : (sgr x ≤ 0) = (x ≤ 0).

sign and norm

Lemma realEsign x : x \is real → x = (-1) ^+ (x < 0)%R × `|x |.

Lemma realNEsign x : x \is real → - x = (-1) ^+ (0 < x)%R × `|x |.

Lemma real_normrEsign (x : R) (xR : x \is real) : `|x | = (-1) ^+ (x < 0)%R × x.

#[deprecated(since="mathcomp 2.3.0", note="use `realEsign` instead")]
Lemma real_mulr_sign_norm x : x \is real → (-1) ^+ (x < 0)%R × `|x | = x.

Lemma real_mulr_Nsign_norm x : x \is real → (-1) ^+ (0 < x)%R × `|x | = - x.

Lemma realEsg x : x \is real → x = sgr x × `|x |.

Lemma normr_sg x : `|sg x | = (x != 0)%:R.

Lemma sgr_norm x : sg `|x | = (x != 0)%:R.

leif

Lemma leif_nat_r m n C : (m %:R ≤ n %:R ?= iff C :> R ) = (m ≤ n ?= iff C)%N.

Lemma leifBLR x y z C : (x - y ≤ z ?= iff C ) = (x ≤ z + y ?= iff C ).

Lemma leifBRL x y z C : (x ≤ y - z ?= iff C ) = (x + z ≤ y ?= iff C ).

Lemma leifD x1 y1 C1 x2 y2 C2 :
    x1 ≤ y1 ?= iff C1 → x2 ≤ y2 ?= iff C2 →
  x1 + x2 ≤ y1 + y2 ?= iff C1 && C2.

Lemma leif_sum (I : finType) (P C : pred I) (E1 E2 : I → R) :
    (∀ i, P i → E1 i ≤ E2 i ?= iff C i ) →
  \sum_(i | P i ) E1 i ≤ \sum_(i | P i ) E2 i ?= iff [∀ (i | P i ), C i ].

Lemma leif_0_sum (I : finType) (P C : pred I) (E : I → R) :
    (∀ i, P i → 0 ≤ E i ?= iff C i ) →
  0 ≤ \sum_(i | P i ) E i ?= iff [∀ (i | P i ), C i ].

Lemma real_leif_norm x : x \is real → x ≤ `|x | ?= iff (0 ≤ x ).

Lemma leif_pM x1 x2 y1 y2 C1 C2 :
    0 ≤ x1 → 0 ≤ x2 → x1 ≤ y1 ?= iff C1 → x2 ≤ y2 ?= iff C2 →
  x1 × x2 ≤ y1 × y2 ?= iff (y1 × y2 == 0) || C1 && C2.

Lemma leif_nM x1 x2 y1 y2 C1 C2 :
    y1 ≤ 0 → y2 ≤ 0 → x1 ≤ y1 ?= iff C1 → x2 ≤ y2 ?= iff C2 →
  y1 × y2 ≤ x1 × x2 ?= iff (x1 × x2 == 0) || C1 && C2.

Lemma leif_pprod (I : finType) (P C : pred I) (E1 E2 : I → R) :
    (∀ i, P i → 0 ≤ E1 i ) →
    (∀ i, P i → E1 i ≤ E2 i ?= iff C i ) →
  let pi E := \prod_(i | P i ) E i in
  pi E1 ≤ pi E2 ?= iff (pi E2 == 0) || [∀ (i | P i ), C i ].

lteif

Lemma subr_lteifr0 C x y : (y - x < 0 ?<= if C ) = (y < x ?<= if C ).

Lemma subr_lteif0r C x y : (0 < y - x ?<= if C ) = (x < y ?<= if C ).

Definition subr_lteif0 := (subr_lteifr0 , subr_lteif0r ).

Lemma lteif01 C : 0 < 1 ?<= if C :> R.

Lemma lteifNl C x y : - x < y ?<= if C = (- y < x ?<= if C ).

Lemma lteifNr C x y : x < - y ?<= if C = (y < - x ?<= if C ).

Lemma lteif0Nr C x : 0 < - x ?<= if C = (x < 0 ?<= if C ).

Lemma lteifNr0 C x : - x < 0 ?<= if C = (0 < x ?<= if C ).

Lemma lteifN2 C : {mono -%R : x y /~ x < y ?<= if C :> R }.

Definition lteif_oppE := (lteif0Nr , lteifNr0 , lteifN2 ).

Lemma lteifD2l C x : {mono +%R x : y z / y < z ?<= if C }.

Lemma lteifD2r C x : {mono +%R ^~ x : y z / y < z ?<= if C }.

Definition lteifD2 := (lteifD2l , lteifD2r ).

Lemma lteifBlDr C x y z : (x - y < z ?<= if C ) = (x < z + y ?<= if C ).

Lemma lteifBrDr C x y z : (x < y - z ?<= if C ) = (x + z < y ?<= if C ).

Definition lteifBDr := (lteifBlDr , lteifBrDr ).

Lemma lteifBlDl C x y z : (x - y < z ?<= if C ) = (x < y + z ?<= if C ).

Lemma lteifBrDl C x y z : (x < y - z ?<= if C ) = (z + x < y ?<= if C ).

Definition lteifBDl := (lteifBlDl , lteifBrDl ).

Lemma lteif_pM2l C x : 0 < x → {mono *%R x : y z / y < z ?<= if C }.

Lemma lteif_pM2r C x : 0 < x → {mono *%R ^~ x : y z / y < z ?<= if C }.

Lemma lteif_nM2l C x : x < 0 → {mono *%R x : y z /~ y < z ?<= if C }.

Lemma lteif_nM2r C x : x < 0 → {mono *%R ^~ x : y z /~ y < z ?<= if C }.

Lemma lteif_nnormr C x y : y < 0 ?<= if ~~ C → (`|x | < y ?<= if C ) = false.

Lemma real_lteifNE x y C : x \is Num.real → y \is Num.real →
  x < y ?<= if ~~ C = ~~ (y < x ?<= if C ).

Lemma real_lteif_norml C x y :
  x \is Num.real →
  (`|x | < y ?<= if C ) = ((- y < x ?<= if C ) && (x < y ?<= if C )).

Lemma real_lteif_normr C x y :
  y \is Num.real →
  (x < `|y | ?<= if C ) = ((x < y ?<= if C ) || (x < - y ?<= if C )).

Lemma real_lteif_distl C x y e :
  x - y \is real →
  (`|x - y | < e ?<= if C ) = (y - e < x ?<= if C ) && (x < y + e ?<= if C ).

Mean inequalities.

Lemma real_leif_mean_square_scaled x y :
  x \is real → y \is real → x × y *+ 2 ≤ x ^+ 2 + y ^+ 2 ?= iff (x == y ).

Lemma real_leif_AGM2_scaled x y :
  x \is real → y \is real → x × y *+ 4 ≤ (x + y ) ^+ 2 ?= iff (x == y ).

Lemma leif_AGM_scaled (I : finType) (A : {pred I }) (E : I → R) (n := #|A |) :
    {in A , ∀ i, 0 ≤ E i *+ n } →
  \prod_(i in A ) (E i *+ n ) ≤ (\sum_(i in A ) E i ) ^+ n
                            ?= iff [∀ i in A , ∀ j in A , E i == E j ].

Polynomial bound.

Implicit Type p : {poly R }.

Lemma poly_disk_bound p b : {ub | ∀ x, `|x | ≤ b → `|p .[x ]| ≤ ub }.

End NumDomainOperationTheory.

#[global] Hint Resolve lerN2 ltrN2 normr_real : core.
#[global] Hint Extern 0 (is_true (_%:R \is real)) ⇒ apply: realn : core.
#[global] Hint Extern 0 (is_true (0 \is real)) ⇒ apply: real0 : core.
#[global] Hint Extern 0 (is_true (1 \is real)) ⇒ apply: real1 : core.

Arguments ler_sqr {R} [x y].
Arguments ltr_sqr {R} [x y].
Arguments signr_inj {R} [x1 x2].
Arguments real_ler_normlP {R x y}.
Arguments real_ltr_normlP {R x y}.

Section NumDomainMonotonyTheoryForReals.
Local Open Scope order_scope.

Variables (R R' : numDomainType) (D : pred R) (f : R → R') (f' : R → nat).
Implicit Types (m n p : nat) (x y z : R) (u v w : R').

Lemma real_mono :
  {homo f : x y / x < y } → {in real &, {mono f : x y / x ≤ y }}.

Lemma real_nmono :
  {homo f : x y /~ x < y } → {in real &, {mono f : x y /~ x ≤ y }}.

Lemma real_mono_in :
    {in D &, {homo f : x y / x < y }} →
  {in [pred x in D | x \is real ] &, {mono f : x y / x ≤ y }}.

Lemma real_nmono_in :
    {in D &, {homo f : x y /~ x < y }} →
  {in [pred x in D | x \is real ] &, {mono f : x y /~ x ≤ y }}.

Lemma realn_mono : {homo f' : x y / x < y >-> (x < y )} →
  {in real &, {mono f' : x y / x ≤ y >-> (x ≤ y )}}.

Lemma realn_nmono : {homo f' : x y / y < x >-> (x < y )} →
  {in real &, {mono f' : x y / y ≤ x >-> (x ≤ y )}}.

Lemma realn_mono_in : {in D &, {homo f' : x y / x < y >-> (x < y )}} →
  {in [pred x in D | x \is real ] &, {mono f' : x y / x ≤ y >-> (x ≤ y )}}.

Lemma realn_nmono_in : {in D &, {homo f' : x y / y < x >-> (x < y )}} →
  {in [pred x in D | x \is real ] &, {mono f' : x y / y ≤ x >-> (x ≤ y )}}.

End NumDomainMonotonyTheoryForReals.

Section FinGroup.

Import GroupScope.

Variables (R : numDomainType) (gT : finGroupType).
Implicit Types G : {group gT }.

Lemma natrG_gt0 G : #|G |%:R > 0 :> R.

Lemma natrG_neq0 G : #|G |%:R != 0 :> R.

Lemma natr_indexg_gt0 G B : #|G : B |%:R > 0 :> R.

Lemma natr_indexg_neq0 G B : #|G : B |%:R != 0 :> R.

End FinGroup.

Section NumFieldTheory.

Variable F : numFieldType.
Implicit Types x y z t : F.

Lemma unitf_gt0 x : 0 < x → x \is a GRing.unit.

Lemma unitf_lt0 x : x < 0 → x \is a GRing.unit.

Lemma lef_pV2 : {in pos &, {mono (@GRing.inv F ) : x y /~ x ≤ y }}.

Lemma lef_nV2 : {in neg &, {mono (@GRing.inv F ) : x y /~ x ≤ y }}.

Lemma ltf_pV2 : {in pos &, {mono (@GRing.inv F ) : x y /~ x < y }}.

Lemma ltf_nV2 : {in neg &, {mono (@GRing.inv F ) : x y /~ x < y }}.

Definition ltef_pV2 := (lef_pV2 , ltf_pV2 ).
Definition ltef_nV2 := (lef_nV2 , ltf_nV2 ).

Lemma invf_pgt : {in pos &, ∀ x y, (x < y ^-1 ) = (y < x ^-1 )}.

Lemma invf_pge : {in pos &, ∀ x y, (x ≤ y ^-1 ) = (y ≤ x ^-1 )}.

Lemma invf_ngt : {in neg &, ∀ x y, (x < y ^-1 ) = (y < x ^-1 )}.

Lemma invf_nge : {in neg &, ∀ x y, (x ≤ y ^-1 ) = (y ≤ x ^-1 )}.

Lemma invf_gt1 x : 0 < x → (1 < x ^-1 ) = (x < 1).

Lemma invf_ge1 x : 0 < x → (1 ≤ x ^-1 ) = (x ≤ 1).

Definition invf_gte1 := (invf_ge1 , invf_gt1 ).

Lemma invf_plt : {in pos &, ∀ x y, (x ^-1 < y ) = (y ^-1 < x )}.

Lemma invf_ple : {in pos &, ∀ x y, (x ^-1 ≤ y ) = (y ^-1 ≤ x )}.

Lemma invf_nlt : {in neg &, ∀ x y, (x ^-1 < y ) = (y ^-1 < x )}.

Lemma invf_nle : {in neg &, ∀ x y, (x ^-1 ≤ y ) = (y ^-1 ≤ x )}.

Lemma invf_le1 x : 0 < x → (x ^-1 ≤ 1) = (1 ≤ x ).

Lemma invf_lt1 x : 0 < x → (x ^-1 < 1) = (1 < x ).

Definition invf_lte1 := (invf_le1 , invf_lt1 ).
Definition invf_cp1 := (invf_gte1 , invf_lte1 ).

These lemma are all combinations of mono(LR|RL) with ler_ [pn]mul2[rl].

Lemma ler_pdivlMr z x y : 0 < z → (x ≤ y / z ) = (x × z ≤ y ).

Lemma ltr_pdivlMr z x y : 0 < z → (x < y / z ) = (x × z < y ).

Definition lter_pdivlMr := (ler_pdivlMr , ltr_pdivlMr ).

Lemma ler_pdivrMr z x y : 0 < z → (y / z ≤ x ) = (y ≤ x × z ).

Lemma ltr_pdivrMr z x y : 0 < z → (y / z < x ) = (y < x × z ).

Definition lter_pdivrMr := (ler_pdivrMr , ltr_pdivrMr ).

Lemma ler_pdivlMl z x y : 0 < z → (x ≤ z ^-1 × y ) = (z × x ≤ y ).

Lemma ltr_pdivlMl z x y : 0 < z → (x < z ^-1 × y ) = (z × x < y ).

Definition lter_pdivlMl := (ler_pdivlMl , ltr_pdivlMl ).

Lemma ler_pdivrMl z x y : 0 < z → (z ^-1 × y ≤ x ) = (y ≤ z × x ).

Lemma ltr_pdivrMl z x y : 0 < z → (z ^-1 × y < x ) = (y < z × x ).

Definition lter_pdivrMl := (ler_pdivrMl , ltr_pdivrMl ).

Lemma ler_ndivlMr z x y : z < 0 → (x ≤ y / z ) = (y ≤ x × z ).

Lemma ltr_ndivlMr z x y : z < 0 → (x < y / z ) = (y < x × z ).

Definition lter_ndivlMr := (ler_ndivlMr , ltr_ndivlMr ).

Lemma ler_ndivrMr z x y : z < 0 → (y / z ≤ x ) = (x × z ≤ y ).

Lemma ltr_ndivrMr z x y : z < 0 → (y / z < x ) = (x × z < y ).

Definition lter_ndivrMr := (ler_ndivrMr , ltr_ndivrMr ).

Lemma ler_ndivlMl z x y : z < 0 → (x ≤ z ^-1 × y ) = (y ≤ z × x ).

Lemma ltr_ndivlMl z x y : z < 0 → (x < z ^-1 × y ) = (y < z × x ).

Definition lter_ndivlMl := (ler_ndivlMl , ltr_ndivlMl ).

Lemma ler_ndivrMl z x y : z < 0 → (z ^-1 × y ≤ x ) = (z × x ≤ y ).

Lemma ltr_ndivrMl z x y : z < 0 → (z ^-1 × y < x ) = (z × x < y ).

Definition lter_ndivrMl := (ler_ndivrMl , ltr_ndivrMl ).

Lemma natf_div m d : (d %| m)%N → (m %/ d )%:R = m %:R / d %:R :> F.

Lemma normfV : {morph (norm : F → F ) : x / x ^-1 }.

Lemma normf_div : {morph (norm : F → F ) : x y / x / y }.

Lemma invr_sg x : (sg x )^-1 = sgr x.

Lemma sgrV x : sgr x ^-1 = sgr x.

Lemma splitr x : x = x / 2%:R + x / 2%:R.

lteif

Lemma lteif_pdivlMr C z x y :
  0 < z → x < y / z ?<= if C = (x × z < y ?<= if C ).

Lemma lteif_pdivrMr C z x y :
  0 < z → y / z < x ?<= if C = (y < x × z ?<= if C ).

Lemma lteif_pdivlMl C z x y :
  0 < z → x < z ^-1 × y ?<= if C = (z × x < y ?<= if C ).

Lemma lteif_pdivrMl C z x y :
  0 < z → z ^-1 × y < x ?<= if C = (y < z × x ?<= if C ).

Lemma lteif_ndivlMr C z x y :
  z < 0 → x < y / z ?<= if C = (y < x × z ?<= if C ).

Lemma lteif_ndivrMr C z x y :
  z < 0 → y / z < x ?<= if C = (x × z < y ?<= if C ).

Lemma lteif_ndivlMl C z x y :
  z < 0 → x < z ^-1 × y ?<= if C = (y < z × x ?<= if C ).

Lemma lteif_ndivrMl C z x y :
  z < 0 → z ^-1 × y < x ?<= if C = (z × x < y ?<= if C ).

Interval midpoint.

Local Notation mid x y := ((x + y) / 2).

Lemma midf_le x y : x ≤ y → (x ≤ mid x y ) × (mid x y ≤ y ).

Lemma midf_lt x y : x < y → (x < mid x y ) × (mid x y < y ).

Definition midf_lte := (midf_le , midf_lt ).

Lemma ler_addgt0Pr x y : reflect (∀ e, e > 0 → x ≤ y + e) (x ≤ y).

Lemma ler_addgt0Pl x y : reflect (∀ e, e > 0 → x ≤ e + y) (x ≤ y).

Lemma lt_le a b : (∀ x, x < a → x < b ) → a ≤ b.

Lemma gt_ge a b : (∀ x, b < x → a < x ) → a ≤ b.

The AGM, unscaled but without the nth root.

Lemma real_leif_mean_square x y :
  x \is real → y \is real → x × y ≤ mid (x ^+ 2) (y ^+ 2) ?= iff (x == y ).

Lemma real_leif_AGM2 x y :
  x \is real → y \is real → x × y ≤ mid x y ^+ 2 ?= iff (x == y ).

Lemma leif_AGM (I : finType) (A : {pred I }) (E : I → F) :
    let n := #|A | in let mu := (\sum_(i in A ) E i ) / n %:R in
    {in A , ∀ i, 0 ≤ E i } →
  \prod_(i in A ) E i ≤ mu ^+ n
                     ?= iff [∀ i in A , ∀ j in A , E i == E j ].

Implicit Type p : {poly F }.
Lemma Cauchy_root_bound p : p != 0 → {b | ∀ x, root p x → `|x | ≤ b }.

Import GroupScope.

Lemma natf_indexg (gT : finGroupType) (G H : {group gT }) :
  H \subset G → #|G : H |%:R = (#|G |%:R / #|H |%:R)%R :> F.

End NumFieldTheory.

Section RealDomainTheory.

Variable R : realDomainType.
Implicit Types x y z t : R.

Lemma num_real x : x \is real.
Hint Resolve num_real : core.

Lemma lerP x y : ler_xor_gt x y (min y x) (min x y) (max y x) (max x y)
                                `|x - y | `|y - x | (x ≤ y) (y < x).

Lemma ltrP x y : ltr_xor_ge x y (min y x) (min x y) (max y x) (max x y)
                                `|x - y | `|y - x | (y ≤ x) (x < y).

Lemma ltrgtP x y :
   comparer x y (min y x) (min x y) (max y x) (max x y)
                 `|x - y | `|y - x | (y == x) (x == y)
                 (x ≥ y) (x ≤ y) (x > y) (x < y) .

Lemma ger0P x : ger0_xor_lt0 x (min 0 x) (min x 0) (max 0 x) (max x 0)
                                `|x | (x < 0) (0 ≤ x).

Lemma ler0P x : ler0_xor_gt0 x (min 0 x) (min x 0) (max 0 x) (max x 0)
                                `|x | (0 < x) (x ≤ 0).

Lemma ltrgt0P x : comparer0 x (min 0 x) (min x 0) (max 0 x) (max x 0)
  `|x | (0 == x) (x == 0) (x ≤ 0) (0 ≤ x) (x < 0) (x > 0).

sign

Lemma mulr_lt0 x y :
  (x × y < 0) = [&& x != 0, y != 0 & (x < 0) (+) (y < 0)].

Lemma neq0_mulr_lt0 x y :
  x != 0 → y != 0 → (x × y < 0) = (x < 0) (+) (y < 0).

Lemma mulr_sign_lt0 (b : bool) x :
  ((-1) ^+ b × x < 0) = (x != 0) && (b (+) (x < 0)%R).

sign & norm

Lemma mulr_sign_norm x : (-1) ^+ (x < 0)%R × `|x | = x.

Lemma mulr_Nsign_norm x : (-1) ^+ (0 < x)%R × `|x | = - x.

Lemma numEsign x : x = (-1) ^+ (x < 0)%R × `|x |.

Lemma numNEsign x : -x = (-1) ^+ (0 < x)%R × `|x |.

Lemma normrEsign x : `|x | = (-1) ^+ (x < 0)%R × x.

End RealDomainTheory.

#[global] Hint Resolve num_real : core.

Section RealDomainOperations.

Notation "[ 'arg' 'min_' ( i < i0 | P ) F ]" :=
    (Order.arg_min (disp := ring_display) i0 (fun i ⇒ P%B) (fun i ⇒ F)) :
   ring_scope.

Notation "[ 'arg' 'min_' ( i < i0 'in' A ) F ]" :=
  [arg min_(i < i0 | i \in A) F] : ring_scope.

Notation "[ 'arg' 'min_' ( i < i0 ) F ]" := [arg min_(i < i0 | true ) F] :
  ring_scope.

Notation "[ 'arg' 'max_' ( i > i0 | P ) F ]" :=
   (Order.arg_max (disp := ring_display) i0 (fun i ⇒ P%B) (fun i ⇒ F)) :
  ring_scope.

Notation "[ 'arg' 'max_' ( i > i0 'in' A ) F ]" :=
    [arg max_(i > i0 | i \in A) F] : ring_scope.

Notation "[ 'arg' 'max_' ( i > i0 ) F ]" := [arg max_(i > i0 | true ) F] :
  ring_scope.

sgr section

Variable R : realDomainType.
Implicit Types x y z t : R.
Let numR_real := @num_real R.
Hint Resolve numR_real : core.

Lemma sgr_cp0 x :
  ((sg x == 1) = (0 < x )) ×
  ((sg x == -1) = (x < 0)) ×
  ((sg x == 0) = (x == 0)).

Variant sgr_val x : R → bool → bool → bool → bool → bool → bool
  → bool → bool → bool → bool → bool → bool → R → Set :=
  | SgrNull of x = 0 : sgr_val x 0 true true true true false false
    true false false true false false 0
  | SgrPos of x > 0 : sgr_val x x false false true false false true
    false false true false false true 1
  | SgrNeg of x < 0 : sgr_val x (- x) false true false false true false
    false true false false true false (-1).

Lemma sgrP x :
  sgr_val x `|x | (0 == x) (x ≤ 0) (0 ≤ x) (x == 0) (x < 0) (0 < x)
                 (0 == sg x) (-1 == sg x) (1 == sg x)
                 (sg x == 0) (sg x == -1) (sg x == 1) (sg x).

Lemma normrEsg x : `|x | = sg x × x.

Lemma numEsg x : x = sg x × `|x |.

#[deprecated(since="mathcomp 2.3.0", note="use `numEsg` instead")]
Lemma mulr_sg_norm x : sg x × `|x | = x.

Lemma sgrM x y : sg (x × y) = sg x × sg y.

Lemma sgrN x : sg (- x) = - sg x.

Lemma sgrX n x : sg (x ^+ n) = (sg x ) ^+ n.

Lemma sgr_smul x y : sg (sg x × y) = sg x × sg y.

Lemma sgr_gt0 x : (sg x > 0) = (x > 0).

Lemma sgr_ge0 x : (sgr x ≥ 0) = (x ≥ 0).

norm section

Lemma ler_norm x : (x ≤ `|x |).

Lemma ler_norml x y : (`|x | ≤ y ) = (- y ≤ x ≤ y ).

Lemma ler_normlP x y : reflect ((- x ≤ y ) × (x ≤ y )) (`|x | ≤ y).
Arguments ler_normlP {x y}.

Lemma eqr_norml x y : (`|x | == y ) = ((x == y ) || (x == -y )) && (0 ≤ y ).

Lemma eqr_norm2 x y : (`|x | == `|y |) = (x == y ) || (x == -y ).

Lemma ltr_norml x y : (`|x | < y ) = (- y < x < y ).

Definition lter_norml := (ler_norml , ltr_norml ).

Lemma ltr_normlP x y : reflect ((-x < y ) × (x < y )) (`|x | < y).
Arguments ltr_normlP {x y}.

Lemma ltr_normlW x y : `|x | < y → x < y.

Lemma ltrNnormlW x y : `|x | < y → - y < x.

Lemma ler_normlW x y : `|x | ≤ y → x ≤ y.

Lemma lerNnormlW x y : `|x | ≤ y → - y ≤ x.

Lemma ler_normr x y : (x ≤ `|y |) = (x ≤ y ) || (x ≤ - y ).

Lemma ltr_normr x y : (x < `|y |) = (x < y ) || (x < - y ).

Definition lter_normr := (ler_normr , ltr_normr ).

Lemma ler_distl x y e : (`|x - y | ≤ e ) = (y - e ≤ x ≤ y + e ).

Lemma ltr_distl x y e : (`|x - y | < e ) = (y - e < x < y + e ).

Definition lter_distl := (ler_distl , ltr_distl ).

Lemma ltr_distlC x y e : (`|x - y | < e ) = (x - e < y < x + e ).

Lemma ler_distlC x y e : (`|x - y | ≤ e ) = (x - e ≤ y ≤ x + e ).

Definition lter_distlC := (ler_distlC , ltr_distlC ).

Lemma ltr_distlDr x y e : `|x - y | < e → x < y + e.

Lemma ler_distlDr x y e : `|x - y | ≤ e → x ≤ y + e.

Lemma ltr_distlCDr x y e : `|x - y | < e → y < x + e.

Lemma ler_distlCDr x y e : `|x - y | ≤ e → y ≤ x + e.

Lemma ltr_distlBl x y e : `|x - y | < e → x - e < y.

Lemma ler_distlBl x y e : `|x - y | ≤ e → x - e ≤ y.

Lemma ltr_distlCBl x y e : `|x - y | < e → y - e < x.

Lemma ler_distlCBl x y e : `|x - y | ≤ e → y - e ≤ x.

Lemma exprn_even_ge0 n x : ~~ odd n → 0 ≤ x ^+ n.

Lemma exprn_even_gt0 n x : ~~ odd n → (0 < x ^+ n ) = (n == 0)%N || (x != 0).

Lemma exprn_even_le0 n x : ~~ odd n → (x ^+ n ≤ 0) = (n != 0) && (x == 0).

Lemma exprn_even_lt0 n x : ~~ odd n → (x ^+ n < 0) = false.

Lemma exprn_odd_ge0 n x : odd n → (0 ≤ x ^+ n ) = (0 ≤ x ).

Lemma exprn_odd_gt0 n x : odd n → (0 < x ^+ n ) = (0 < x ).

Lemma exprn_odd_le0 n x : odd n → (x ^+ n ≤ 0) = (x ≤ 0).

Lemma exprn_odd_lt0 n x : odd n → (x ^+ n < 0) = (x < 0).

lteif

Lemma lteif_norml C x y :
  (`|x | < y ?<= if C ) = (- y < x ?<= if C ) && (x < y ?<= if C ).

Lemma lteif_normr C x y :
  (x < `|y | ?<= if C ) = (x < y ?<= if C ) || (x < - y ?<= if C ).

Lemma lteif_distl C x y e :
  (`|x - y | < e ?<= if C ) = (y - e < x ?<= if C ) && (x < y + e ?<= if C ).

Special lemmas for squares.

Lemma sqr_ge0 x : 0 ≤ x ^+ 2.

Lemma sqr_norm_eq1 x : (x ^+ 2 == 1) = (`|x | == 1).

Lemma leif_mean_square_scaled x y :
x × y *+ 2 ≤ x ^+ 2 + y ^+ 2 ?= iff (x == y ).

Lemma leif_AGM2_scaled x y : x × y *+ 4 ≤ (x + y ) ^+ 2 ?= iff (x == y ).

Section MinMax.

Lemma oppr_max : {morph -%R : x y / max x y >-> min x y : R }.

Lemma oppr_min : {morph -%R : x y / min x y >-> max x y : R }.

Lemma addr_minl : @left_distributive R R +%R min.

Lemma addr_minr : @right_distributive R R +%R min.

Lemma addr_maxl : @left_distributive R R +%R max.

Lemma addr_maxr : @right_distributive R R +%R max.

Lemma minr_nMr x y z : x ≤ 0 → x × min y z = max (x × y) (x × z).

Lemma maxr_nMr x y z : x ≤ 0 → x × max y z = min (x × y) (x × z).

Lemma minr_nMl x y z : x ≤ 0 → min y z × x = max (y × x) (z × x).

Lemma maxr_nMl x y z : x ≤ 0 → max y z × x = min (y × x) (z × x).

Lemma maxrN x : max x (- x) = `|x |.
Lemma maxNr x : max (- x) x = `|x |.
Lemma minrN x : min x (- x) = - `|x |.
Lemma minNr x : min (- x) x = - `|x |.

End MinMax.

Section PolyBounds.

Variable p : {poly R }.

Lemma poly_itv_bound a b : {ub | ∀ x, a ≤ x ≤ b → `|p .[x ]| ≤ ub }.

Lemma monic_Cauchy_bound : p \is monic → {b | ∀ x, x ≥ b → p .[x ] > 0}.

End PolyBounds.

End RealDomainOperations.

Section RealField.

Variables (F : realFieldType) (x y : F).

Lemma leif_mean_square : x × y ≤ (x ^+ 2 + y ^+ 2) / 2 ?= iff (x == y ).

Lemma leif_AGM2 : x × y ≤ ((x + y ) / 2)^+ 2 ?= iff (x == y ).

End RealField.

Section RealClosedFieldTheory.

Variable R : rcfType.
Implicit Types a x y : R.

Lemma poly_ivt : real_closed_axiom R.

Square Root theory

Lemma sqrtr_ge0 a : 0 ≤ sqrt a.
Hint Resolve sqrtr_ge0 : core.

Lemma sqr_sqrtr a : 0 ≤ a → sqrt a ^+ 2 = a.

Lemma ler0_sqrtr a : a ≤ 0 → sqrt a = 0.

Lemma ltr0_sqrtr a : a < 0 → sqrt a = 0.

Variant sqrtr_spec a : R → bool → bool → R → Type :=
| IsNoSqrtr of a < 0 : sqrtr_spec a a false true 0
| IsSqrtr b of 0 ≤ b : sqrtr_spec a (b ^+ 2) true false b.

Lemma sqrtrP a : sqrtr_spec a a (0 ≤ a) (a < 0) (sqrt a).

Lemma sqrtr_sqr a : sqrt (a ^+ 2) = `|a |.

Lemma sqrtrM a b : 0 ≤ a → sqrt (a × b) = sqrt a × sqrt b.

Lemma sqrtr0 : sqrt 0 = 0 :> R.

Lemma sqrtr1 : sqrt 1 = 1 :> R.

Lemma sqrtr_eq0 a : (sqrt a == 0) = (a ≤ 0).

Lemma sqrtr_gt0 a : (0 < sqrt a ) = (0 < a ).

Lemma eqr_sqrt a b : 0 ≤ a → 0 ≤ b → (sqrt a == sqrt b ) = (a == b ).

Lemma ler_wsqrtr : {homo @sqrt R : a b / a ≤ b }.

Lemma ler_psqrt : {in @nneg R &, {mono sqrt : a b / a ≤ b }}.

Lemma ler_sqrt a b : 0 ≤ b → (sqrt a ≤ sqrt b ) = (a ≤ b ).

Lemma ltr_sqrt a b : 0 < b → (sqrt a < sqrt b ) = (a < b ).

Lemma sqrtrV x : 0 ≤ x → sqrt (x ^-1) = (sqrt x )^-1.

End RealClosedFieldTheory.

Notation "z ^*" := (conj_op z) : ring_scope.
Notation "'i" := imaginary (at level 0) : ring_scope.

Section ClosedFieldTheory.

Variable C : numClosedFieldType.
Implicit Types a x y z : C.

Definition normCK : ∀ x, `|x | ^+ 2 = x × x ^* := normCK.

Definition sqrCi : 'i ^+ 2 = -1 :> C := sqrCi.

Lemma mulCii : 'i × 'i = -1 :> C.

Lemma conjCK : involutive (@conj_op C).

Let Re2 z := z + z ^*.
Definition nnegIm z := (0 ≤ 'i × (z ^* - z )).
Definition argCle y z := nnegIm z ==> nnegIm y && (Re2 z ≤ Re2 y ).

Variant rootC_spec n (x : C) : Type :=
RootCspec (y : C) of if (n > 0)%N then y ^+ n = x else y = 0
& ∀ z, (n > 0)%N → z ^+ n = x → argCle y z.

Fact rootC_subproof n x : rootC_spec n x.

Definition nthroot n x := let: RootCspec y _ _ := rootC_subproof n x in y.
Notation "n .-root" := (nthroot n) : ring_scope.
Notation sqrtC := 2.-root.

Fact Re_lock : unit.
Fact Im_lock : unit.
Definition Re z := locked_with Re_lock ((z + z ^*) / 2%:R).
Definition Im z := locked_with Im_lock ('i × (z ^* - z ) / 2%:R).
Notation "'Re z" := (Re z) : ring_scope.
Notation "'Im z" := (Im z) : ring_scope.

Lemma ReE z : 'Re z = (z + z ^*) / 2%:R.
Lemma ImE z : 'Im z = 'i × (z ^* - z ) / 2%:R.

Let nz2 : 2 != 0 :> C.

Lemma normCKC x : `|x | ^+ 2 = x ^* × x.

Lemma mul_conjC_ge0 x : 0 ≤ x × x ^*.

Lemma mul_conjC_gt0 x : (0 < x × x ^* ) = (x != 0).

Lemma mul_conjC_eq0 x : (x × x ^* == 0) = (x == 0).

Lemma conjC_ge0 x : (0 ≤ x ^* ) = (0 ≤ x ).

Lemma conjC_nat n : (n %:R )^* = n %:R :> C.
Lemma conjC0 : 0^* = 0 :> C.
Lemma conjC1 : 1^* = 1 :> C.
Lemma conjCN1 : (- 1)^* = - 1 :> C.
Lemma conjC_eq0 x : (x ^* == 0) = (x == 0).

Lemma invC_norm x : x ^-1 = `|x | ^- 2 × x ^*.

Real number subset.

Lemma CrealE x : (x \is real ) = (x ^* == x ).

Lemma CrealP {x} : reflect (x ^* = x) (x \is real).

Lemma conj_Creal x : x \is real → x ^* = x.

Lemma conj_normC z : `|z |^* = `|z |.

Lemma CrealJ : {mono (@conj_op C ) : x / x \is Num.real }.

Lemma geC0_conj x : 0 ≤ x → x ^* = x.

Lemma geC0_unit_exp x n : 0 ≤ x → (x ^+ n .+1 == 1) = (x == 1).

Elementary properties of roots.

Ltac case_rootC := rewrite /nthroot; case: (rootC_subproof _ _).

Lemma root0C x : 0.-root x = 0.

Lemma rootCK n : (n > 0)%N → cancel n .-root (fun x ⇒ x ^+ n).

Lemma root1C x : 1.-root x = x.

Lemma rootC0 n : n .-root 0 = 0.

Lemma rootC_inj n : (n > 0)%N → injective n .-root.

Lemma eqr_rootC n : (n > 0)%N → {mono n .-root : x y / x == y }.

Lemma rootC_eq0 n x : (n > 0)%N → (n .-root x == 0) = (x == 0).

Rectangular coordinates.

Lemma nonRealCi : ('i : C ) \isn't real.

Lemma neq0Ci : 'i != 0 :> C.

Lemma normCi : `|'i | = 1 :> C.

Lemma invCi : 'i ^-1 = - 'i :> C.

Lemma conjCi : 'i ^* = - 'i :> C.

Lemma Crect x : x = 'Re x + 'i × 'Im x.

Lemma eqCP x y : x = y ↔ ('Re x = 'Re y ) ∧ ('Im x = 'Im y ).

Lemma eqC x y : (x == y ) = ('Re x == 'Re y ) && ('Im x == 'Im y ).

Lemma Creal_Re x : 'Re x \is real.

Lemma Creal_Im x : 'Im x \is real.
Hint Resolve Creal_Re Creal_Im : core.

Fact Re_is_additive : additive Re.
#[export ]
HB.instance Definition _ := GRing.isAdditive.Build C C Re Re_is_additive.

Fact Im_is_additive : additive Im.
#[export ]
HB.instance Definition _ := GRing.isAdditive.Build C C Im Im_is_additive.

Lemma Creal_ImP z : reflect ('Im z = 0) (z \is real).

Lemma Creal_ReP z : reflect ('Re z = z) (z \in real).

Lemma ReMl : {in real , ∀ x, {morph Re : z / x × z }}.

Lemma ReMr : {in real , ∀ x, {morph Re : z / z × x }}.

Lemma ImMl : {in real , ∀ x, {morph Im : z / x × z }}.

Lemma ImMr : {in real , ∀ x, {morph Im : z / z × x }}.

Lemma Re_i : 'Re 'i = 0.

Lemma Im_i : 'Im 'i = 1.

Lemma Re_conj z : 'Re z ^* = 'Re z.

Lemma Im_conj z : 'Im z ^* = - 'Im z.

Lemma Re_rect : {in real &, ∀ x y, 'Re (x + 'i × y ) = x }.

Lemma Im_rect : {in real &, ∀ x y, 'Im (x + 'i × y ) = y }.

Lemma conjC_rect : {in real &, ∀ x y, (x + 'i × y )^* = x - 'i × y }.

Lemma addC_rect x1 y1 x2 y2 :
  (x1 + 'i × y1 ) + (x2 + 'i × y2 ) = x1 + x2 + 'i × (y1 + y2 ).

Lemma oppC_rect x y : - (x + 'i × y ) = - x + 'i × (- y ).

Lemma subC_rect x1 y1 x2 y2 :
  (x1 + 'i × y1 ) - (x2 + 'i × y2 ) = x1 - x2 + 'i × (y1 - y2 ).

Lemma mulC_rect x1 y1 x2 y2 : (x1 + 'i × y1 ) × (x2 + 'i × y2 ) =
                              x1 × x2 - y1 × y2 + 'i × (x1 × y2 + x2 × y1 ).

Lemma ImM x y : 'Im (x × y ) = 'Re x × 'Im y + 'Re y × 'Im x.

Lemma ImMil x : 'Im ('i × x ) = 'Re x.

Lemma ReMil x : 'Re ('i × x ) = - 'Im x.

Lemma ReMir x : 'Re (x × 'i ) = - 'Im x.

Lemma ImMir x : 'Im (x × 'i ) = 'Re x.

Lemma ReM x y : 'Re (x × y ) = 'Re x × 'Re y - 'Im x × 'Im y.

Lemma normC2_rect :
  {in real &, ∀ x y, `|x + 'i × y | ^+ 2 = x ^+ 2 + y ^+ 2}.

Lemma normC2_Re_Im z : `|z | ^+ 2 = 'Re z ^+ 2 + 'Im z ^+ 2.

Lemma invC_Crect x y : (x + 'i × y )^-1 = (x ^* - 'i × y ^*) / `|x + 'i × y | ^+ 2.

Lemma invC_rect :
  {in real &, ∀ x y, (x + 'i × y )^-1 = (x - 'i × y ) / (x ^+ 2 + y ^+ 2)}.

Lemma ImV x : 'Im x ^-1 = - 'Im x / `|x | ^+ 2.

Lemma ReV x : 'Re x ^-1 = 'Re x / `|x | ^+ 2.

Lemma rectC_mulr x y z : (x + 'i × y ) × z = x × z + 'i × (y × z ).

Lemma rectC_mull x y z : z × (x + 'i × y ) = z × x + 'i × (z × y ).

Lemma divC_Crect x1 y1 x2 y2 :
  (x1 + 'i × y1 ) / (x2 + 'i × y2 ) =
  (x1 × x2 ^* + y1 × y2 ^* + 'i × (x2 ^* × y1 - x1 × y2 ^*)) /
    `|x2 + 'i × y2 | ^+ 2.

Lemma divC_rect x1 y1 x2 y2 :
     x1 \is real → y1 \is real → x2 \is real → y2 \is real →
  (x1 + 'i × y1 ) / (x2 + 'i × y2 ) =
  (x1 × x2 + y1 × y2 + 'i × (x2 × y1 - x1 × y2 )) /
    (x2 ^+ 2 + y2 ^+ 2).

Lemma Im_div x y : 'Im (x / y ) = ('Re y × 'Im x - 'Re x × 'Im y ) / `|y | ^+ 2.

Lemma Re_div x y : 'Re (x / y ) = ('Re x × 'Re y + 'Im x × 'Im y ) / `|y | ^+ 2.

Lemma leif_normC_Re_Creal z : `|'Re z | ≤ `|z | ?= iff (z \is real ).

Lemma leif_Re_Creal z : 'Re z ≤ `|z | ?= iff (0 ≤ z ).

Equality from polar coordinates, for the upper plane.

Lemma eqC_semipolar x y :
`|x | = `|y | → 'Re x = 'Re y → 0 ≤ 'Im x × 'Im y → x = y.

Nth roots.

Let argCleP y z :
reflect (0 ≤ 'Im z → 0 ≤ 'Im y ∧ 'Re z ≤ 'Re y) (argCle y z).

Lemma rootC_Re_max n x y :
(n > 0)%N → y ^+ n = x → 0 ≤ 'Im y → 'Re y ≤ 'Re (n .-root x ).

Let neg_unity_root n : (n > 1)%N → exists2 w : C , w ^+ n = 1 & 'Re w < 0.

Lemma Im_rootC_ge0 n x : (n > 1)%N → 0 ≤ 'Im (n .-root x ).

Lemma rootC_lt0 n x : (1 < n)%N → (n .-root x < 0) = false.

Lemma rootC_ge0 n x : (n > 0)%N → (0 ≤ n .-root x ) = (0 ≤ x ).

Lemma rootC_gt0 n x : (n > 0)%N → (n .-root x > 0) = (x > 0).

Lemma rootC_le0 n x : (1 < n)%N → (n .-root x ≤ 0) = (x == 0).

Lemma ler_rootCl n : (n > 0)%N → {in Num.nneg , {mono n .-root : x y / x ≤ y }}.

Lemma ler_rootC n : (n > 0)%N → {in Num.nneg &, {mono n .-root : x y / x ≤ y }}.

Lemma ltr_rootCl n : (n > 0)%N → {in Num.nneg , {mono n .-root : x y / x < y }}.

Lemma ltr_rootC n : (n > 0)%N → {in Num.nneg &, {mono n .-root : x y / x < y }}.

Lemma exprCK n x : (0 < n)%N → 0 ≤ x → n .-root (x ^+ n ) = x.

Lemma norm_rootC n x : `|n .-root x | = n .-root `|x |.

Lemma rootCX n x k : (n > 0)%N → 0 ≤ x → n .-root (x ^+ k ) = n .-root x ^+ k.

Lemma rootC1 n : (n > 0)%N → n .-root 1 = 1.

Lemma rootCpX n x k : (k > 0)%N → 0 ≤ x → n .-root (x ^+ k ) = n .-root x ^+ k.

Lemma rootCV n x : 0 ≤ x → n .-root x ^-1 = (n .-root x )^-1.

Lemma rootC_eq1 n x : (n > 0)%N → (n .-root x == 1) = (x == 1).

Lemma rootC_ge1 n x : (n > 0)%N → (n .-root x ≥ 1) = (x ≥ 1).

Lemma rootC_gt1 n x : (n > 0)%N → (n .-root x > 1) = (x > 1).

Lemma rootC_le1 n x : (n > 0)%N → 0 ≤ x → (n .-root x ≤ 1) = (x ≤ 1).

Lemma rootC_lt1 n x : (n > 0)%N → 0 ≤ x → (n .-root x < 1) = (x < 1).

Lemma rootCMl n x z : 0 ≤ x → n .-root (x × z ) = n .-root x × n .-root z.

Lemma rootCMr n x z : 0 ≤ x → n .-root (z × x ) = n .-root z × n .-root x.

Lemma imaginaryCE : 'i = sqrtC (-1).

More properties of n.-root will be established in cyclotomic.v. The proper form of the Arithmetic - Geometric Mean inequality.

Lemma leif_rootC_AGM (I : finType) (A : {pred I }) (n := #|A |) E :
    {in A , ∀ i, 0 ≤ E i } →
  n .-root (\prod_(i in A ) E i ) ≤ (\sum_(i in A ) E i ) / n %:R
                             ?= iff [∀ i in A , ∀ j in A , E i == E j ].

Square root.

Lemma sqrtC0 : sqrtC 0 = 0.
Lemma sqrtC1 : sqrtC 1 = 1.
Lemma sqrtCK x : sqrtC x ^+ 2 = x.
Lemma sqrCK x : 0 ≤ x → sqrtC (x ^+ 2) = x.

Lemma sqrtC_ge0 x : (0 ≤ sqrtC x ) = (0 ≤ x ).
Lemma sqrtC_eq0 x : (sqrtC x == 0) = (x == 0).
Lemma sqrtC_gt0 x : (sqrtC x > 0) = (x > 0).
Lemma sqrtC_lt0 x : (sqrtC x < 0) = false.
Lemma sqrtC_le0 x : (sqrtC x ≤ 0) = (x == 0).

Lemma ler_sqrtC : {in Num.nneg &, {mono sqrtC : x y / x ≤ y }}.
Lemma ltr_sqrtC : {in Num.nneg &, {mono sqrtC : x y / x < y }}.
Lemma eqr_sqrtC : {mono sqrtC : x y / x == y }.
Lemma sqrtC_inj : injective sqrtC.
Lemma sqrtCM : {in Num.nneg &, {morph sqrtC : x y / x × y }}.

Lemma sqrtC_real x : 0 ≤ x → sqrtC x \in Num.real.

Lemma sqrCK_P x : reflect (sqrtC (x ^+ 2) = x) ((0 ≤ 'Im x ) && ~~ (x < 0)).

Lemma normC_def x : `|x | = sqrtC (x × x ^* ).

Lemma norm_conjC x : `|x ^*| = `|x |.

Lemma normC_rect :
{in real &, ∀ x y, `|x + 'i × y | = sqrtC (x ^+ 2 + y ^+ 2)}.

Lemma normC_Re_Im z : `|z | = sqrtC ('Re z ^+ 2 + 'Im z ^+ 2).

Norm sum (in)equalities.

Lemma normCDeq x y :
    `|x + y | = `|x | + `|y | →
  {t : C | `|t | == 1 & (x , y ) = (`|x | × t , `|y | × t )}.

Lemma normC_sum_eq (I : finType) (P : pred I) (F : I → C) :
     `|\sum_(i | P i ) F i | = \sum_(i | P i ) `|F i | →
   {t : C | `|t | == 1 & ∀ i, P i → F i = `|F i | × t }.

Lemma normC_sum_eq1 (I : finType) (P : pred I) (F : I → C) :
    `|\sum_(i | P i ) F i | = (\sum_(i | P i ) `|F i |) →
     (∀ i, P i → `|F i | = 1) →
   {t : C | `|t | == 1 & ∀ i, P i → F i = t }.

Lemma normC_sum_upper (I : finType) (P : pred I) (F G : I → C) :
     (∀ i, P i → `|F i | ≤ G i ) →
     \sum_(i | P i ) F i = \sum_(i | P i ) G i →
   ∀ i, P i → F i = G i.

Lemma normCBeq x y :
  `|x - y | = `|x | - `|y | → {t | `|t | == 1 & (x , y ) = (`|x | × t , `|y | × t )}.

End ClosedFieldTheory.

Notation "n .-root" := (@nthroot _ n).
Notation sqrtC := 2.-root.
Notation "'i" := imaginary : ring_scope.
Notation "'Re z" := (Re z) : ring_scope.
Notation "'Im z" := (Im z) : ring_scope.

Arguments conjCK {C} x.
Arguments sqrCK {C} [x] le0x.
Arguments sqrCK_P {C x}.

#[global] Hint Extern 0 (is_true (in_mem ('Re _) _)) ⇒
  solve [apply: Creal_Re] : core.
#[global] Hint Extern 0 (is_true (in_mem ('Im _) _)) ⇒
  solve [apply: Creal_Im] : core.

Module Export Pdeg2.

Module NumClosed.

Section Pdeg2NumClosed.

Variables (F : numClosedFieldType) (p : {poly F }).

Hypothesis degp : size p = 3.

Let a := p `_2.
Let b := p `_1.
Let c := p `_0.

Let delta := b ^+ 2 - 4 × a × c.

Let r1 := (- b - sqrtC delta ) / (2 × a ).
Let r2 := (- b + sqrtC delta ) / (2 × a ).

Lemma deg2_poly_factor : p = a *: ('X - r1 %:P ) × ('X - r2 %:P ).

Lemma deg2_poly_root1 : root p r1.

Lemma deg2_poly_root2 : root p r2.

End Pdeg2NumClosed.

End NumClosed.

Module NumClosedMonic.

Export FieldMonic.

Section Pdeg2NumClosedMonic.

Variables (F : numClosedFieldType) (p : {poly F }).

Hypothesis degp : size p = 3.
Hypothesis monicp : p \is monic.

Let a := p `_2.
Let b := p `_1.
Let c := p `_0.

Let delta := b ^+ 2 - 4 × c.

Let r1 := (- b - sqrtC delta ) / 2.
Let r2 := (- b + sqrtC delta ) / 2.

Lemma deg2_poly_factor : p = ('X - r1 %:P ) × ('X - r2 %:P ).

Lemma deg2_poly_root1 : root p r1.

Lemma deg2_poly_root2 : root p r2.

End Pdeg2NumClosedMonic.
End NumClosedMonic.

Module Real.

Section Pdeg2Real.

Variable F : realFieldType.

Section Pdeg2RealConvex.

Variable p : {poly F }.
Hypothesis degp : size p = 3.

Let a := p `_2.
Let b := p `_1.
Let c := p `_0.

Hypothesis age0 : 0 ≤ a.

Let delta := b ^+ 2 - 4 × a × c.

Let pneq0 : p != 0.
Let aneq0 : a != 0.
Let agt0 : 0 < a.
Let a4gt0 : 0 < 4 × a.

Lemma deg2_poly_min x : p .[- b / (2 × a )] ≤ p .[x ].

Lemma deg2_poly_minE : p .[- b / (2 × a )] = - delta / (4 × a ).

Lemma deg2_poly_gt0 : reflect (∀ x, 0 < p .[x ]) (delta < 0).

Lemma deg2_poly_ge0 : reflect (∀ x, 0 ≤ p .[x ]) (delta ≤ 0).

End Pdeg2RealConvex.

Section Pdeg2RealConcave.

Variable p : {poly F }.
Hypothesis degp : size p = 3.

Let a := p `_2.
Let b := p `_1.
Let c := p `_0.

Hypothesis ale0 : a ≤ 0.

Let delta := b ^+ 2 - 4 × a × c.

Let degpN : size (- p) = 3.
Let b2a : - (- p )`_1 / (2 × (- p )`_2 ) = - b / (2 × a ).
Let deltaN : (- p )`_1 ^+ 2 - 4 × (- p )`_2 × (- p )`_0 = delta.

Lemma deg2_poly_max x : p .[x ] ≤ p .[- b / (2 × a )].

Lemma deg2_poly_maxE : p .[- b / (2 × a )] = - delta / (4 × a ).

Lemma deg2_poly_lt0 : reflect (∀ x, p .[x ] < 0) (delta < 0).

Lemma deg2_poly_le0 : reflect (∀ x, p .[x ] ≤ 0) (delta ≤ 0).

End Pdeg2RealConcave.
End Pdeg2Real.

Section Pdeg2RealClosed.

Variable F : rcfType.

Section Pdeg2RealClosedConvex.

Variable p : {poly F }.
Hypothesis degp : size p = 3.

Let a := p `_2.
Let b := p `_1.
Let c := p `_0.

Let pneq0 : p != 0.
Let aneq0 : a != 0.
Let sqa2 : 4 × a ^+ 2 = (2 × a ) ^+ 2.
Let nz2 : 2 != 0 :> F.

Let delta := b ^+ 2 - 4 × a × c.

Let r1 := (- b - sqrt delta ) / (2 × a ).
Let r2 := (- b + sqrt delta ) / (2 × a ).

Lemma deg2_poly_factor : 0 ≤ delta → p = a *: ('X - r1 %:P ) × ('X - r2 %:P ).

Lemma deg2_poly_root1 : 0 ≤ delta → root p r1.

Lemma deg2_poly_root2 : 0 ≤ delta → root p r2.

Lemma deg2_poly_noroot : reflect (∀ x, ~~ root p x) (delta < 0).

Hypothesis age0 : 0 ≤ a.

Let agt0 : 0 < a.
Let a2gt0 : 0 < 2 × a.
Let a4gt0 : 0 < 4 × a.
Let aa4gt0 : 0 < 4 × a × a.

Let xb4 x : (x + b / (2 × a )) ^+ 2 × (4 × a × a ) = (x × (2 × a ) + b ) ^+ 2.

Lemma deg2_poly_gt0l x : x < r1 → 0 < p .[x ].

Lemma deg2_poly_gt0r x : r2 < x → 0 < p .[x ].

Lemma deg2_poly_lt0m x : r1 < x < r2 → p .[x ] < 0.

Lemma deg2_poly_ge0l x : x ≤ r1 → 0 ≤ p .[x ].

Lemma deg2_poly_ge0r x : r2 ≤ x → 0 ≤ p .[x ].

Lemma deg2_poly_le0m x : 0 ≤ delta → r1 ≤ x ≤ r2 → p .[x ] ≤ 0.

End Pdeg2RealClosedConvex.

Section Pdeg2RealClosedConcave.

Variable p : {poly F }.
Hypothesis degp : size p = 3.

Let a := p `_2.
Let b := p `_1.
Let c := p `_0.

Let delta := b ^+ 2 - 4 × a × c.

Let r1 := (- b + sqrt delta ) / (2 × a ).
Let r2 := (- b - sqrt delta ) / (2 × a ).

Hypothesis ale0 : a ≤ 0.

Let degpN : size (- p) = 3.
Let aNge0 : 0 ≤ (- p )`_2.
Let deltaN : (- p )`_1 ^+ 2 - 4 × (- p )`_2 × (- p )`_0 = delta.
Let r1N : (- (- p )`_1 - sqrt delta ) / (2 × (- p )`_2 ) = r1.
Let r2N : (- (- p )`_1 + sqrt delta ) / (2 × (- p )`_2 ) = r2.

Lemma deg2_poly_lt0l x : x < r1 → p .[x ] < 0.

Lemma deg2_poly_lt0r x : r2 < x → p .[x ] < 0.

Lemma deg2_poly_gt0m x : r1 < x < r2 → 0 < p .[x ].

Lemma deg2_poly_le0l x : x ≤ r1 → p .[x ] ≤ 0.

Lemma deg2_poly_le0r x : r2 ≤ x → p .[x ] ≤ 0.

Lemma deg2_poly_ge0m x : 0 ≤ delta → r1 ≤ x ≤ r2 → 0 ≤ p .[x ].

End Pdeg2RealClosedConcave.
End Pdeg2RealClosed.
End Real.

Module RealMonic.

Import Real.
Export FieldMonic.

Section Pdeg2RealMonic.

Variable F : realFieldType.

Variable p : {poly F }.
Hypothesis degp : size p = 3.
Hypothesis monicp : p \is monic.

Let a := p `_2.
Let b := p `_1.
Let c := p `_0.

Let delta := b ^+ 2 - 4 × c.

Let a1 : a = 1.
Let a2 : 2 × a = 2.
Let a4 : 4 × a = 4.

Lemma deg2_poly_min x : p .[- b / 2] ≤ p .[x ].

Let deltam : delta = b ^+ 2 - 4 × a × c.

Lemma deg2_poly_minE : p .[- b / 2] = - delta / 4.

Lemma deg2_poly_gt0 : reflect (∀ x, 0 < p .[x ]) (delta < 0).

Lemma deg2_poly_ge0 : reflect (∀ x, 0 ≤ p .[x ]) (delta ≤ 0).

End Pdeg2RealMonic.

Section Pdeg2RealClosedMonic.

Variables (F : rcfType) (p : {poly F }).
Hypothesis degp : size p = 3.
Hypothesis monicp : p \is monic.

Let a := p `_2.
Let b := p `_1.
Let c := p `_0.

Let a1 : a = 1.

Let delta := b ^+ 2 - 4 × c.

Let deltam : delta = b ^+ 2 - 4 × a × c.

Let r1 := (- b - sqrt delta ) / 2.
Let r2 := (- b + sqrt delta ) / 2.

Let nz2 : 2 != 0 :> F.

Lemma deg2_poly_factor : 0 ≤ delta → p = ('X - r1 %:P ) × ('X - r2 %:P ).

Lemma deg2_poly_root1 : 0 ≤ delta → root p r1.

Lemma deg2_poly_root2 : 0 ≤ delta → root p r2.

Lemma deg2_poly_noroot : reflect (∀ x, ~~ root p x) (delta < 0).

Lemma deg2_poly_gt0l x : x < r1 → 0 < p .[x ].

Lemma deg2_poly_gt0r x : r2 < x → 0 < p .[x ].

Lemma deg2_poly_lt0m x : r1 < x < r2 → p .[x ] < 0.

Lemma deg2_poly_ge0l x : x ≤ r1 → 0 ≤ p .[x ].

Lemma deg2_poly_ge0r x : r2 ≤ x → 0 ≤ p .[x ].

Lemma deg2_poly_le0m x : 0 ≤ delta → r1 ≤ x ≤ r2 → p .[x ] ≤ 0.

End Pdeg2RealClosedMonic.
End RealMonic.
End Pdeg2.

Section Degle2PolyRealConvex.

Variable (F : realFieldType) (p : {poly F }).
Hypothesis degp : (size p ≤ 3)%N.

Let a := p `_2.
Let b := p `_1.
Let c := p `_0.

Let delta := b ^+ 2 - 4 × a × c.

Lemma deg_le2_poly_delta_ge0 : 0 ≤ a → (∀ x, 0 ≤ p .[x ]) → delta ≤ 0.

End Degle2PolyRealConvex.

Section Degle2PolyRealConcave.

Variable (F : realFieldType) (p : {poly F }).
Hypothesis degp : (size p ≤ 3)%N.

Let a := p `_2.
Let b := p `_1.
Let c := p `_0.

Let delta := b ^+ 2 - 4 × a × c.

Lemma deg_le2_poly_delta_le0 : a ≤ 0 → (∀ x, p .[x ] ≤ 0) → delta ≤ 0.

End Degle2PolyRealConcave.

Section Degle2PolyRealClosedConvex.

Variable (F : rcfType) (p : {poly F }).
Hypothesis degp : (size p ≤ 3)%N.

Let a := p `_2.
Let b := p `_1.
Let c := p `_0.

Let delta := b ^+ 2 - 4 × a × c.

Lemma deg_le2_poly_ge0 : (∀ x, 0 ≤ p .[x ]) → delta ≤ 0.

End Degle2PolyRealClosedConvex.

Section Degle2PolyRealClosedConcave.

Variable (F : rcfType) (p : {poly F }).
Hypothesis degp : (size p ≤ 3)%N.

Let a := p `_2.
Let b := p `_1.
Let c := p `_0.

Let delta := b ^+ 2 - 4 × a × c.

Lemma deg_le2_poly_le0 : (∀ x, p .[x ] ≤ 0) → delta ≤ 0.

End Degle2PolyRealClosedConcave.

End Theory.

  Local Notation "x <= y" := (Rle x y) : ring_scope.
  Local Notation "x < y" := (Rlt x y) : ring_scope.
  Local Notation "`| x |" := (norm x) : ring_scope.

  Lemma ltrr x : x < x = false.

  Lemma ge0_def x : (0 ≤ x ) = (`|x | == x ).

  Lemma subr_ge0 x y : (0 ≤ x - y ) = (y ≤ x ).

  Lemma subr_gt0 x y : (0 < y - x ) = (x < y ).

  Lemma lt_trans : transitive Rlt.

  Lemma le01 : 0 ≤ 1.

  Lemma lt01 : 0 < 1.

  Lemma ltW x y : x < y → x ≤ y.

  Lemma lerr x : x ≤ x.

  Lemma le_def' x y : (x ≤ y ) = (x == y ) || (x < y ).

  Lemma le_trans : transitive Rle.

  Lemma normrMn x n : `|x *+ n | = `|x | *+ n.

  Lemma normrN1 : `|-1| = 1 :> R.

  Lemma normrN x : `|- x | = `|x |.

  Lemma le_total : Order.POrder_isTotal ring_display R.

  Local Notation le := Rle.
  Local Notation lt := Rlt.

  Local Notation "x <= y" := (le x y) : ring_scope.
  Local Notation "x < y" := (lt x y) : ring_scope.
  Local Notation "`| x |" := (norm x) : ring_scope.

  Let le0N x : (0 ≤ - x ) = (x ≤ 0).
  Let leN_total x : 0 ≤ x ∨ 0 ≤ - x.

  Let le00 : 0 ≤ 0.

  Fact lt0_add x y : 0 < x → 0 < y → 0 < x + y.

  Fact eq0_norm x : `|x | = 0 → x = 0.

  Fact le_def x y : (x ≤ y ) = (`|y - x | == y - x ).

  Fact normM : {morph norm : x y / x × y }.

  Fact le_normD x y : `|x + y | ≤ `|x | + `|y |.

  Fact le_total : total le.

  Local Notation le := Rle.
  Local Notation lt := Rlt.

  Local Notation "x < y" := (lt x y) : ring_scope.
  Local Notation "x <= y" := (le x y) : ring_scope.
  Local Notation "`| x |" := (norm x) : ring_scope.

  Fact lt0N x : (- x < 0) = (0 < x ).
  Let leN_total x : 0 ≤ x ∨ 0 ≤ - x.

  Let le00 : (0 ≤ 0).

  Fact sub_ge0 x y : (0 ≤ y - x ) = (x ≤ y ).

  Fact le0_add x y : 0 ≤ x → 0 ≤ y → 0 ≤ x + y.

  Fact le0_mul x y : 0 ≤ x → 0 ≤ y → 0 ≤ x × y.

  Fact normM : {morph norm : x y / x × y }.

  Fact le_normD x y : `|x + y | ≤ `|x | + `|y |.

  Fact eq0_norm x : `|x | = 0 → x = 0.

  Fact le_def' x y : (x ≤ y ) = (`|y - x | == y - x ).

  Fact lt_def x y : (x < y ) = (y != x ) && (x ≤ y ).

  Fact le_total : total le.

Module Exports. End Exports.

End Num.
Export Num.Exports.

Export Num.Syntax Num.PredInstances.