Library mathcomp.algebra.ssrnum

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  
 Distributed under the terms of CeCILL-B.                                  *)

From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice.
From mathcomp Require Import div fintype path bigop finset fingroup.
From mathcomp Require Import ssralg poly.

This file defines some classes to manipulate number structures, i.e structures with an order and a norm

NumDomain (Integral domain with an order and a norm)

NumMixin == the mixin that provides an order and a norm over a ring and their characteristic properties. numDomainType == interface for a num integral domain. NumDomainType T m == packs the num mixin into a numberDomainType. The carrier T must have a integral domain structure. [numDomainType of T for S ] == T-clone of the numDomainType structure S. [numDomainType of T] == clone of a canonical numDomainType structure on T.

NumField (Field with an order and a norm)

numFieldType == interface for a num field. [numFieldType of T] == clone of a canonical numFieldType structure on T

NumClosedField (Closed Field with an order and a norm)

numClosedFieldType == interface for a num closed field. [numClosedFieldType of T] == clone of a canonical numClosedFieldType structure on T

RealDomain (Num domain where all elements are positive or negative)

realDomainType == interface for a real integral domain. RealDomainType T r == packs the real axiom r into a realDomainType. The carrier T must have a num domain structure. [realDomainType of T for S ] == T-clone of the realDomainType structure S. [realDomainType of T] == clone of a canonical realDomainType structure on T.

RealField (Num Field where all elements are positive or negative)

realFieldType == interface for a real field. [realFieldType of T] == clone of a canonical realFieldType structure on T

ArchiField (A Real Field with the archimedean axiom)

archiFieldType == interface for an archimedean field. ArchiFieldType T r == packs the archimeadean axiom r into an archiFieldType. The carrier T must have a real field type structure. [archiFieldType of T for S ] == T-clone of the archiFieldType structure S. [archiFieldType of T] == clone of a canonical archiFieldType structure on T

RealClosedField (Real Field with the real closed axiom)

rcfType == interface for a real closed field. RcfType T r == packs the real closed axiom r into a rcfType. The carrier T must have a real field type structure. [rcfType of T] == clone of a canonical realClosedFieldType structure on T. [rcfType of T for S ] == T-clone of the realClosedFieldType structure S.

NumClosedField (Partially ordered Closed Field with conjugation)

numClosedFieldType == interface for a closed field with conj. NumClosedFieldType T r == packs the real closed axiom r into a numClosedFieldType. The carrier T must have a closed field type structure. [numClosedFieldType of T] == clone of a canonical numClosedFieldType structure on T [numClosedFieldType of T for S ] == T-clone of the realClosedFieldType structure S.
Over these structures, we have the following operations `|x| == norm of x. x <= y <=> x is less than or equal to y (:= '|y - x| == y - x). x < y <=> x is less than y (:= (x <= y) && (x != y)). x <= y ?= iff C <-> x is less than y, or equal iff C is true. 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.real <=> x is real (:= x >= 0 or x < 0). Num.min x y == minimum of x y Num.max x y == maximum of x y Num.bound x == in archimedean fields, and upper bound for x, i.e., and n such that `|x| < n%:R. 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).
There are now three distinct uses of the symbols <, <=, > and >=: 0-ary, unary (prefix) and binary (infix). 0. <%R, <=%R, >%R, >=%R stand respectively for lt, le, gt and ge. 1. (< x), (<= x), (> x), (>= x) stand respectively for (gt x), (ge x), (lt x), (le x). So (< x) is a predicate characterizing elements smaller than x. 2. (x < y), (x <= y), ... mean what they are expected to. These convention are compatible with haskell's, where ((< y) x) = (x < y) = ((<) x y), except that we write <%R instead of (<).
  • 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
[arg minr(i < i0 | P) M] == a value i : T minimizing M : R, subject to the condition P (i may appear in P and M), and provided P holds for i0. [arg maxr(i > i0 | P) M] == a value i maximizing M subject to P and provided P holds for i0. [arg minr(i < i0 in A) M] == an i \in A minimizing M if i0 \in A. [arg maxr(i > i0 in A) M] == an i \in A maximizing M if i0 \in A. [arg minr(i < i0) M] == an i : T minimizing M, given i0 : T. [arg maxr(i > i0) M] == an i : T maximizing M, given i0 : T.

Set Implicit Arguments.

Local Open Scope ring_scope.
Import GRing.Theory.

Reserved Notation "<= y" (at level 35).
Reserved Notation ">= y" (at level 35).
Reserved Notation "< y" (at level 35).
Reserved Notation "> y" (at level 35).
Reserved Notation "<= y :> T" (at level 35, y at next level).
Reserved Notation ">= y :> T" (at level 35, y at next level).
Reserved Notation "< y :> T" (at level 35, y at next level).
Reserved Notation "> y :> T" (at level 35, y at next level).

Module Num.

Principal mixin; further classes add axioms rather than operations.
Record mixin_of (R : ringType) := Mixin {
  norm_op : R R;
  le_op : rel R;
  lt_op : rel R;
  _ : x y, le_op (norm_op (x + y)) (norm_op x + norm_op y);
  _ : x y, lt_op 0 x lt_op 0 y lt_op 0 (x + y);
  _ : x, norm_op x = 0 x = 0;
  _ : x y, le_op 0 x le_op 0 y le_op x y || le_op y x;
  _ : {morph norm_op : x y / x × y};
  _ : x y, (le_op x y) = (norm_op (y - x) == y - x);
  _ : x y, (lt_op x y) = (y != x) && (le_op x y)
}.


Base interface.
Module NumDomain.

Section ClassDef.

Record class_of T := Class {
  base : GRing.IntegralDomain.class_of T;
  mixin : mixin_of (ring_for T base)
}.
Structure type := Pack {sort; _ : class_of sort}.
Variables (T : Type) (cT : type).
Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Let xT := let: Pack T _ := cT in T.
Notation xclass := (class : class_of xT).

Definition clone c of phant_id class c := @Pack T c.
Definition pack b0 (m0 : mixin_of (ring_for T b0)) :=
  fun bT b & phant_id (GRing.IntegralDomain.class bT) b
  fun m & phant_id m0 mPack (@Class T b m).

Definition eqType := @Equality.Pack cT xclass.
Definition choiceType := @Choice.Pack cT xclass.
Definition zmodType := @GRing.Zmodule.Pack cT xclass.
Definition ringType := @GRing.Ring.Pack cT xclass.
Definition comRingType := @GRing.ComRing.Pack cT xclass.
Definition unitRingType := @GRing.UnitRing.Pack cT xclass.
Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass.
Definition idomainType := @GRing.IntegralDomain.Pack cT xclass.

End ClassDef.

Module Exports.
Coercion base : class_of >-> GRing.IntegralDomain.class_of.
Coercion mixin : class_of >-> mixin_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Canonical eqType.
Coercion choiceType : type >-> Choice.type.
Canonical choiceType.
Coercion zmodType : type >-> GRing.Zmodule.type.
Canonical zmodType.
Coercion ringType : type >-> GRing.Ring.type.
Canonical ringType.
Coercion comRingType : type >-> GRing.ComRing.type.
Canonical comRingType.
Coercion unitRingType : type >-> GRing.UnitRing.type.
Canonical unitRingType.
Coercion comUnitRingType : type >-> GRing.ComUnitRing.type.
Canonical comUnitRingType.
Coercion idomainType : type >-> GRing.IntegralDomain.type.
Canonical idomainType.
Notation numDomainType := type.
Notation NumMixin := Mixin.
Notation NumDomainType T m := (@pack T _ m _ _ id _ id).
Notation "[ 'numDomainType' 'of' T 'for' cT ]" := (@clone T cT _ idfun)
  (at level 0, format "[ 'numDomainType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'numDomainType' 'of' T ]" := (@clone T _ _ id)
  (at level 0, format "[ 'numDomainType' 'of' T ]") : form_scope.
End Exports.

End NumDomain.
Import NumDomain.Exports.

Module Import Def. Section Def.
Import NumDomain.
Context {R : type}.
Implicit Types (x y : R) (C : bool).

Definition normr : R R := norm_op (class R).
Definition ler : rel R := le_op (class R).
Definition ltr : rel R := lt_op (class R).

Definition ger : simpl_rel R := [rel x y | y x].
Definition gtr : simpl_rel R := [rel x y | y < x].
Definition lerif x y C : Prop := ((x y) × ((x == y) = C))%type.
Definition sgr x : R := if x == 0 then 0 else if x < 0 then -1 else 1.
Definition minr x y : R := if x y then x else y.
Definition maxr x y : R := if y x then x else y.

Definition Rpos : qualifier 0 R := [qualify x : R | 0 < x].
Definition Rneg : qualifier 0 R := [qualify x : R | x < 0].
Definition Rnneg : qualifier 0 R := [qualify x : R | 0 x].
Definition Rreal : qualifier 0 R := [qualify x : R | (0 x) || (x 0)].
End Def. End Def.

Shorter qualified names, when Num.Def is not imported.
Notation norm := normr.
Notation le := ler.
Notation lt := ltr.
Notation ge := ger.
Notation gt := gtr.
Notation sg := sgr.
Notation max := maxr.
Notation min := minr.
Notation pos := Rpos.
Notation neg := Rneg.
Notation nneg := Rnneg.
Notation real := Rreal.

Module Keys. Section Keys.
Variable R : numDomainType.
Fact Rpos_key : pred_key (@pos R).
Definition Rpos_keyed := KeyedQualifier Rpos_key.
Fact Rneg_key : pred_key (@real R).
Definition Rneg_keyed := KeyedQualifier Rneg_key.
Fact Rnneg_key : pred_key (@nneg R).
Definition Rnneg_keyed := KeyedQualifier Rnneg_key.
Fact Rreal_key : pred_key (@real R).
Definition Rreal_keyed := KeyedQualifier Rreal_key.
Definition ler_of_leif x y C (le_xy : @lerif R x y C) := le_xy.1 : le x y.
End Keys. End Keys.

(Exported) symbolic syntax.
Module Import Syntax.
Import Def Keys.

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

Notation "<%R" := lt : ring_scope.
Notation ">%R" := gt : ring_scope.
Notation "<=%R" := le : ring_scope.
Notation ">=%R" := ge : ring_scope.
Notation "<?=%R" := lerif : ring_scope.

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

Notation "<= y" := (ge y) : ring_scope.
Notation "<= y :> T" := ( (y : T)) : ring_scope.
Notation ">= y" := (le y) : ring_scope.
Notation ">= y :> T" := ( (y : T)) : ring_scope.

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

Coercion ler_of_leif : lerif >-> is_true.

Canonical Rpos_keyed.
Canonical Rneg_keyed.
Canonical Rnneg_keyed.
Canonical Rreal_keyed.

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.
Module NumField.

Section ClassDef.

Record class_of R :=
  Class { base : GRing.Field.class_of R; mixin : mixin_of (ring_for R base) }.
Definition base2 R (c : class_of R) := NumDomain.Class (mixin c).

Structure type := Pack {sort; _ : class_of sort}.
Variables (T : Type) (cT : type).
Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Let xT := let: Pack T _ := cT in T.
Notation xclass := (class : class_of xT).

Definition pack :=
  fun bT b & phant_id (GRing.Field.class bT) (b : GRing.Field.class_of T) ⇒
  fun mT m & phant_id (NumDomain.class mT) (@NumDomain.Class T b m) ⇒
  Pack (@Class T b m).

Definition eqType := @Equality.Pack cT xclass.
Definition choiceType := @Choice.Pack cT xclass.
Definition zmodType := @GRing.Zmodule.Pack cT xclass.
Definition ringType := @GRing.Ring.Pack cT xclass.
Definition comRingType := @GRing.ComRing.Pack cT xclass.
Definition unitRingType := @GRing.UnitRing.Pack cT xclass.
Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass.
Definition idomainType := @GRing.IntegralDomain.Pack cT xclass.
Definition numDomainType := @NumDomain.Pack cT xclass.
Definition fieldType := @GRing.Field.Pack cT xclass.
Definition join_numDomainType := @NumDomain.Pack fieldType xclass.

End ClassDef.

Module Exports.
Coercion base : class_of >-> GRing.Field.class_of.
Coercion base2 : class_of >-> NumDomain.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Canonical eqType.
Coercion choiceType : type >-> Choice.type.
Canonical choiceType.
Coercion zmodType : type >-> GRing.Zmodule.type.
Canonical zmodType.
Coercion ringType : type >-> GRing.Ring.type.
Canonical ringType.
Coercion comRingType : type >-> GRing.ComRing.type.
Canonical comRingType.
Coercion unitRingType : type >-> GRing.UnitRing.type.
Canonical unitRingType.
Coercion comUnitRingType : type >-> GRing.ComUnitRing.type.
Canonical comUnitRingType.
Coercion idomainType : type >-> GRing.IntegralDomain.type.
Canonical idomainType.
Coercion numDomainType : type >-> NumDomain.type.
Canonical numDomainType.
Coercion fieldType : type >-> GRing.Field.type.
Canonical fieldType.
Canonical join_numDomainType.
Notation numFieldType := type.
Notation "[ 'numFieldType' 'of' T ]" := (@pack T _ _ id _ _ id)
  (at level 0, format "[ 'numFieldType' 'of' T ]") : form_scope.
End Exports.

End NumField.
Import NumField.Exports.

Module ClosedField.

Section ClassDef.

Record imaginary_mixin_of (R : numDomainType) := ImaginaryMixin {
  imaginary : R;
  conj_op : {rmorphism R R};
  _ : imaginary ^+ 2 = - 1;
  _ : x, x × conj_op x = `|x| ^+ 2;
}.

Record class_of R := Class {
  base : GRing.ClosedField.class_of R;
  mixin : mixin_of (ring_for R base);
  conj_mixin : imaginary_mixin_of (num_for R (NumDomain.Class mixin))
}.
Definition base2 R (c : class_of R) := NumField.Class (mixin c).

Structure type := Pack {sort; _ : class_of sort}.
Variables (T : Type) (cT : type).
Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Let xT := let: Pack T _ := cT in T.
Notation xclass := (class : class_of xT).

Definition pack :=
  fun bT b & phant_id (GRing.ClosedField.class bT)
                      (b : GRing.ClosedField.class_of T) ⇒
  fun mT m & phant_id (NumField.class mT) (@NumField.Class T b m) ⇒
  fun mcPack (@Class T b m mc).
Definition clone := fun b & phant_id class (b : class_of T) ⇒ Pack b.

Definition eqType := @Equality.Pack cT xclass.
Definition choiceType := @Choice.Pack cT xclass.
Definition zmodType := @GRing.Zmodule.Pack cT xclass.
Definition ringType := @GRing.Ring.Pack cT xclass.
Definition comRingType := @GRing.ComRing.Pack cT xclass.
Definition unitRingType := @GRing.UnitRing.Pack cT xclass.
Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass.
Definition idomainType := @GRing.IntegralDomain.Pack cT xclass.
Definition numDomainType := @NumDomain.Pack cT xclass.
Definition fieldType := @GRing.Field.Pack cT xclass.
Definition numFieldType := @NumField.Pack cT xclass.
Definition decFieldType := @GRing.DecidableField.Pack cT xclass.
Definition closedFieldType := @GRing.ClosedField.Pack cT xclass.
Definition join_dec_numDomainType := @NumDomain.Pack decFieldType xclass.
Definition join_dec_numFieldType := @NumField.Pack decFieldType xclass.
Definition join_numDomainType := @NumDomain.Pack closedFieldType xclass.
Definition join_numFieldType := @NumField.Pack closedFieldType xclass.

End ClassDef.

Module Exports.
Coercion base : class_of >-> GRing.ClosedField.class_of.
Coercion base2 : class_of >-> NumField.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Canonical eqType.
Coercion choiceType : type >-> Choice.type.
Canonical choiceType.
Coercion zmodType : type >-> GRing.Zmodule.type.
Canonical zmodType.
Coercion ringType : type >-> GRing.Ring.type.
Canonical ringType.
Coercion comRingType : type >-> GRing.ComRing.type.
Canonical comRingType.
Coercion unitRingType : type >-> GRing.UnitRing.type.
Canonical unitRingType.
Coercion comUnitRingType : type >-> GRing.ComUnitRing.type.
Canonical comUnitRingType.
Coercion idomainType : type >-> GRing.IntegralDomain.type.
Canonical idomainType.
Coercion numDomainType : type >-> NumDomain.type.
Canonical numDomainType.
Coercion fieldType : type >-> GRing.Field.type.
Canonical fieldType.
Coercion decFieldType : type >-> GRing.DecidableField.type.
Canonical decFieldType.
Coercion numFieldType : type >-> NumField.type.
Canonical numFieldType.
Coercion closedFieldType : type >-> GRing.ClosedField.type.
Canonical closedFieldType.
Canonical join_dec_numDomainType.
Canonical join_dec_numFieldType.
Canonical join_numDomainType.
Canonical join_numFieldType.
Notation numClosedFieldType := type.
Notation NumClosedFieldType T m := (@pack T _ _ id _ _ id m).
Notation "[ 'numClosedFieldType' 'of' T 'for' cT ]" := (@clone T cT _ id)
  (at level 0, format "[ 'numClosedFieldType' 'of' T 'for' cT ]") :
                                                         form_scope.
Notation "[ 'numClosedFieldType' 'of' T ]" := (@clone T _ _ id)
  (at level 0, format "[ 'numClosedFieldType' 'of' T ]") : form_scope.
End Exports.

End ClosedField.
Import ClosedField.Exports.

Module RealDomain.

Section ClassDef.

Record class_of R :=
  Class {base : NumDomain.class_of R; _ : @real_axiom (num_for R base)}.

Structure type := Pack {sort; _ : class_of sort}.
Variables (T : Type) (cT : type).
Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Let xT := let: Pack T _ := cT in T.
Notation xclass := (class : class_of xT).

Definition clone c of phant_id class c := @Pack T c.
Definition pack b0 (m0 : real_axiom (num_for T b0)) :=
  fun bT b & phant_id (NumDomain.class bT) b
  fun m & phant_id m0 mPack (@Class T b m).

Definition eqType := @Equality.Pack cT xclass.
Definition choiceType := @Choice.Pack cT xclass.
Definition zmodType := @GRing.Zmodule.Pack cT xclass.
Definition ringType := @GRing.Ring.Pack cT xclass.
Definition comRingType := @GRing.ComRing.Pack cT xclass.
Definition unitRingType := @GRing.UnitRing.Pack cT xclass.
Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass.
Definition idomainType := @GRing.IntegralDomain.Pack cT xclass.
Definition numDomainType := @NumDomain.Pack cT xclass.

End ClassDef.

Module Exports.
Coercion base : class_of >-> NumDomain.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Canonical eqType.
Coercion choiceType : type >-> Choice.type.
Canonical choiceType.
Coercion zmodType : type >-> GRing.Zmodule.type.
Canonical zmodType.
Coercion ringType : type >-> GRing.Ring.type.
Canonical ringType.
Coercion comRingType : type >-> GRing.ComRing.type.
Canonical comRingType.
Coercion unitRingType : type >-> GRing.UnitRing.type.
Canonical unitRingType.
Coercion comUnitRingType : type >-> GRing.ComUnitRing.type.
Canonical comUnitRingType.
Coercion idomainType : type >-> GRing.IntegralDomain.type.
Canonical idomainType.
Coercion numDomainType : type >-> NumDomain.type.
Canonical numDomainType.
Notation realDomainType := type.
Notation RealDomainType T m := (@pack T _ m _ _ id _ id).
Notation "[ 'realDomainType' 'of' T 'for' cT ]" := (@clone T cT _ idfun)
  (at level 0, format "[ 'realDomainType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'realDomainType' 'of' T ]" := (@clone T _ _ id)
  (at level 0, format "[ 'realDomainType' 'of' T ]") : form_scope.
End Exports.

End RealDomain.
Import RealDomain.Exports.

Module RealField.

Section ClassDef.

Record class_of R :=
  Class { base : NumField.class_of R; mixin : real_axiom (num_for R base) }.
Definition base2 R (c : class_of R) := RealDomain.Class (@mixin R c).

Structure type := Pack {sort; _ : class_of sort}.
Variables (T : Type) (cT : type).
Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Let xT := let: Pack T _ := cT in T.
Notation xclass := (class : class_of xT).

Definition pack :=
  fun bT b & phant_id (NumField.class bT) (b : NumField.class_of T) ⇒
  fun mT m & phant_id (RealDomain.class mT) (@RealDomain.Class T b m) ⇒
  Pack (@Class T b m).

Definition eqType := @Equality.Pack cT xclass.
Definition choiceType := @Choice.Pack cT xclass.
Definition zmodType := @GRing.Zmodule.Pack cT xclass.
Definition ringType := @GRing.Ring.Pack cT xclass.
Definition comRingType := @GRing.ComRing.Pack cT xclass.
Definition unitRingType := @GRing.UnitRing.Pack cT xclass.
Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass.
Definition idomainType := @GRing.IntegralDomain.Pack cT xclass.
Definition numDomainType := @NumDomain.Pack cT xclass.
Definition realDomainType := @RealDomain.Pack cT xclass.
Definition fieldType := @GRing.Field.Pack cT xclass.
Definition numFieldType := @NumField.Pack cT xclass.
Definition join_fieldType := @GRing.Field.Pack realDomainType xclass.
Definition join_numFieldType := @NumField.Pack realDomainType xclass.

End ClassDef.

Module Exports.
Coercion base : class_of >-> NumField.class_of.
Coercion base2 : class_of >-> RealDomain.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Canonical eqType.
Coercion choiceType : type >-> Choice.type.
Canonical choiceType.
Coercion zmodType : type >-> GRing.Zmodule.type.
Canonical zmodType.
Coercion ringType : type >-> GRing.Ring.type.
Canonical ringType.
Coercion comRingType : type >-> GRing.ComRing.type.
Canonical comRingType.
Coercion unitRingType : type >-> GRing.UnitRing.type.
Canonical unitRingType.
Coercion comUnitRingType : type >-> GRing.ComUnitRing.type.
Canonical comUnitRingType.
Coercion idomainType : type >-> GRing.IntegralDomain.type.
Canonical idomainType.
Coercion numDomainType : type >-> NumDomain.type.
Canonical numDomainType.
Coercion realDomainType : type >-> RealDomain.type.
Canonical realDomainType.
Coercion fieldType : type >-> GRing.Field.type.
Canonical fieldType.
Coercion numFieldType : type >-> NumField.type.
Canonical numFieldType.
Canonical join_fieldType.
Canonical join_numFieldType.
Notation realFieldType := type.
Notation "[ 'realFieldType' 'of' T ]" := (@pack T _ _ id _ _ id)
  (at level 0, format "[ 'realFieldType' 'of' T ]") : form_scope.
End Exports.

End RealField.
Import RealField.Exports.

Module ArchimedeanField.

Section ClassDef.

Record class_of R :=
  Class { base : RealField.class_of R; _ : archimedean_axiom (num_for R base) }.

Structure type := Pack {sort; _ : class_of sort}.
Variables (T : Type) (cT : type).
Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Let xT := let: Pack T _ := cT in T.
Notation xclass := (class : class_of xT).

Definition clone c of phant_id class c := @Pack T c.
Definition pack b0 (m0 : archimedean_axiom (num_for T b0)) :=
  fun bT b & phant_id (RealField.class bT) b
  fun m & phant_id m0 mPack (@Class T b m).

Definition eqType := @Equality.Pack cT xclass.
Definition choiceType := @Choice.Pack cT xclass.
Definition zmodType := @GRing.Zmodule.Pack cT xclass.
Definition ringType := @GRing.Ring.Pack cT xclass.
Definition comRingType := @GRing.ComRing.Pack cT xclass.
Definition unitRingType := @GRing.UnitRing.Pack cT xclass.
Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass.
Definition idomainType := @GRing.IntegralDomain.Pack cT xclass.
Definition numDomainType := @NumDomain.Pack cT xclass.
Definition realDomainType := @RealDomain.Pack cT xclass.
Definition fieldType := @GRing.Field.Pack cT xclass.
Definition numFieldType := @NumField.Pack cT xclass.
Definition realFieldType := @RealField.Pack cT xclass.

End ClassDef.

Module Exports.
Coercion base : class_of >-> RealField.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Canonical eqType.
Coercion choiceType : type >-> Choice.type.
Canonical choiceType.
Coercion zmodType : type >-> GRing.Zmodule.type.
Canonical zmodType.
Coercion ringType : type >-> GRing.Ring.type.
Canonical ringType.
Coercion comRingType : type >-> GRing.ComRing.type.
Canonical comRingType.
Coercion unitRingType : type >-> GRing.UnitRing.type.
Canonical unitRingType.
Coercion comUnitRingType : type >-> GRing.ComUnitRing.type.
Canonical comUnitRingType.
Coercion idomainType : type >-> GRing.IntegralDomain.type.
Canonical idomainType.
Coercion numDomainType : type >-> NumDomain.type.
Canonical numDomainType.
Coercion realDomainType : type >-> RealDomain.type.
Canonical realDomainType.
Coercion fieldType : type >-> GRing.Field.type.
Canonical fieldType.
Coercion numFieldType : type >-> NumField.type.
Canonical numFieldType.
Coercion realFieldType : type >-> RealField.type.
Canonical realFieldType.
Notation archiFieldType := type.
Notation ArchiFieldType T m := (@pack T _ m _ _ id _ id).
Notation "[ 'archiFieldType' 'of' T 'for' cT ]" := (@clone T cT _ idfun)
  (at level 0, format "[ 'archiFieldType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'archiFieldType' 'of' T ]" := (@clone T _ _ id)
  (at level 0, format "[ 'archiFieldType' 'of' T ]") : form_scope.
End Exports.

End ArchimedeanField.
Import ArchimedeanField.Exports.

Module RealClosedField.

Section ClassDef.

Record class_of R :=
  Class { base : RealField.class_of R; _ : real_closed_axiom (num_for R base) }.

Structure type := Pack {sort; _ : class_of sort}.
Variables (T : Type) (cT : type).
Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Let xT := let: Pack T _ := cT in T.
Notation xclass := (class : class_of xT).

Definition clone c of phant_id class c := @Pack T c.
Definition pack b0 (m0 : real_closed_axiom (num_for T b0)) :=
  fun bT b & phant_id (RealField.class bT) b
  fun m & phant_id m0 mPack (@Class T b m).

Definition eqType := @Equality.Pack cT xclass.
Definition choiceType := @Choice.Pack cT xclass.
Definition zmodType := @GRing.Zmodule.Pack cT xclass.
Definition ringType := @GRing.Ring.Pack cT xclass.
Definition comRingType := @GRing.ComRing.Pack cT xclass.
Definition unitRingType := @GRing.UnitRing.Pack cT xclass.
Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass.
Definition idomainType := @GRing.IntegralDomain.Pack cT xclass.
Definition numDomainType := @NumDomain.Pack cT xclass.
Definition realDomainType := @RealDomain.Pack cT xclass.
Definition fieldType := @GRing.Field.Pack cT xclass.
Definition numFieldType := @NumField.Pack cT xclass.
Definition realFieldType := @RealField.Pack cT xclass.

End ClassDef.

Module Exports.
Coercion base : class_of >-> RealField.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Canonical eqType.
Coercion choiceType : type >-> Choice.type.
Canonical choiceType.
Coercion zmodType : type >-> GRing.Zmodule.type.
Canonical zmodType.
Coercion ringType : type >-> GRing.Ring.type.
Canonical ringType.
Coercion comRingType : type >-> GRing.ComRing.type.
Canonical comRingType.
Coercion unitRingType : type >-> GRing.UnitRing.type.
Canonical unitRingType.
Coercion comUnitRingType : type >-> GRing.ComUnitRing.type.
Canonical comUnitRingType.
Coercion idomainType : type >-> GRing.IntegralDomain.type.
Canonical idomainType.
Coercion numDomainType : type >-> NumDomain.type.
Canonical numDomainType.
Coercion realDomainType : type >-> RealDomain.type.
Canonical realDomainType.
Coercion fieldType : type >-> GRing.Field.type.
Canonical fieldType.
Coercion numFieldType : type >-> NumField.type.
Canonical numFieldType.
Coercion realFieldType : type >-> RealField.type.
Canonical realFieldType.
Notation rcfType := Num.RealClosedField.type.
Notation RcfType T m := (@pack T _ m _ _ id _ id).
Notation "[ 'rcfType' 'of' T 'for' cT ]" := (@clone T cT _ idfun)
  (at level 0, format "[ 'rcfType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'rcfType' 'of' T ]" := (@clone T _ _ id)
  (at level 0, format "[ 'rcfType' 'of' T ]") : form_scope.
End Exports.

End RealClosedField.
Import RealClosedField.Exports.

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

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

Lemmas from the signature

Lemma normr0_eq0 x : `|x| = 0 x = 0.

Lemma ler_norm_add x y : `|x + y| `|x| + `|y|.

Lemma addr_gt0 x y : 0 < x 0 < y 0 < x + y.

Lemma ger_leVge x y : 0 x 0 y (x y) || (y x).

Lemma normrM : {morph norm : x y / x × y : R}.

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

Lemma ltr_def x y : (x < y) = (y != x) && (x y).

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 ltrW x y : x < y x y.

Lemma lerr x : x x.

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).
Canonical pos_mulrPred := MulrPred pos_divr_closed.
Canonical pos_divrPred := DivrPred pos_divr_closed.

Fact nneg_divr_closed : divr_closed (@nneg R).
Canonical nneg_mulrPred := MulrPred nneg_divr_closed.
Canonical nneg_divrPred := DivrPred nneg_divr_closed.

Fact nneg_addr_closed : addr_closed (@nneg R).
Canonical nneg_addrPred := AddrPred nneg_addr_closed.
Canonical nneg_semiringPred := SemiringPred nneg_divr_closed.

Fact real_oppr_closed : oppr_closed (@real R).
Canonical real_opprPred := OpprPred real_oppr_closed.

Fact real_addr_closed : addr_closed (@real R).
Canonical real_addrPred := AddrPred real_addr_closed.
Canonical real_zmodPred := ZmodPred real_oppr_closed.

Fact real_divr_closed : divr_closed (@real R).
Canonical real_mulrPred := MulrPred real_divr_closed.
Canonical real_smulrPred := SmulrPred real_divr_closed.
Canonical real_divrPred := DivrPred real_divr_closed.
Canonical real_sdivrPred := SdivrPred real_divr_closed.
Canonical real_semiringPred := SemiringPred real_divr_closed.
Canonical real_subringPred := SubringPred real_divr_closed.
Canonical real_divringPred := DivringPred real_divr_closed.

End Domain.

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

Fact archi_bound_subproof (R : archiFieldType) : archimedean_axiom R.

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.

End Internals.

Module PredInstances.

Canonical pos_mulrPred.
Canonical pos_divrPred.

Canonical nneg_addrPred.
Canonical nneg_mulrPred.
Canonical nneg_divrPred.
Canonical nneg_semiringPred.

Canonical real_addrPred.
Canonical real_opprPred.
Canonical real_zmodPred.
Canonical real_mulrPred.
Canonical real_smulrPred.
Canonical real_divrPred.
Canonical real_sdivrPred.
Canonical real_semiringPred.
Canonical real_subringPred.
Canonical real_divringPred.

End PredInstances.

Module Import ExtraDef.

Definition archi_bound {R} x := sval (sigW (@archi_bound_subproof R x)).

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

End ExtraDef.

Notation bound := archi_bound.
Notation sqrt := sqrtr.

Module Theory.

Section NumIntegralDomainTheory.

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

Lemmas from the signature (reexported from internals).

Definition ler_norm_add x y : `|x + y| `|x| + `|y| := ler_norm_add x y.
Definition addr_gt0 x y : 0 < x 0 < y 0 < x + y := @addr_gt0 R x y.
Definition normr0_eq0 x : `|x| = 0 x = 0 := @normr0_eq0 R x.
Definition ger_leVge x y : 0 x 0 y (x y) || (y x) :=
  @ger_leVge R x y.
Definition normrM : {morph normr : x y / x × y : R} := @normrM R.
Definition ler_def x y : (x y) = (`|y - x| == y - x) := @ler_def R x y.
Definition ltr_def x y : (x < y) = (y != x) && (x y) := @ltr_def R x y.

Predicate and relation definitions.

Lemma gerE x y : ge x y = (y x).
Lemma gtrE x y : gt x y = (y < x).
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 realE x : (x \is real) = (0 x) || (x 0).

General properties of <= and <

Lemma lerr x : x x.
Lemma ltrr x : x < x = false.
Lemma ltrW x y : x < y x y.
Hint Resolve lerr ltrr ltrW : core.

Lemma ltr_neqAle x y : (x < y) = (x != y) && (x y).

Lemma ler_eqVlt x y : (x y) = (x == y) || (x < y).

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 gtr_eqF x y : y < x x == y = false.

Lemma ltr_eqF x y : x < y x == y = false.

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 Resolve ler01 ltr01 ler0n : core.
Lemma ltr0Sn n : 0 < n.+1%:R :> R.
Lemma ltr0n n : (0 < n%:R :> R) = (0 < n)%N.
Hint Resolve ltr0Sn : core.

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

Lemma char_num : [char 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 normr0 : `|0| = 0 :> R.
Lemma normr1 : `|1| = 1 :> R.
Lemma normr_nat n : `|n%:R| = n%:R :> R.
Lemma normrMn x n : `|x *+ n| = `|x| *+ n.

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) : x / x \is a GRing.unit}.

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

Lemma normr0P {x} : reflect (`|x| = 0) (x == 0).

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

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

Lemma normrN x : `|- x| = `|x|.

Lemma distrC x y : `|x - y| = `|y - x|.

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

Lemma normr_id x : `|`|x| | = `|x|.

Lemma normr_ge0 x : 0 `|x|.
Hint Resolve normr_ge0 : core.

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

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

Comparision 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).
Lemma subr_lt0 x y : (y - x < 0) = (y < x).

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

Ordered ring properties.

Lemma ler_asym : antisymmetric (<=%R : rel R).

Lemma eqr_le x y : (x == y) = (x y x).

Lemma ltr_trans : transitive (@ltr R).

Lemma ler_lt_trans y x z : x y y < z x < z.

Lemma ltr_le_trans y x z : x < y y z x < z.

Lemma ler_trans : transitive (@ler R).

Definition lter01 := (ler01, ltr01).
Definition lterr := (lerr, ltrr).

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

Lemma lerifP x y C : reflect (x y ?= iff C) (if C then x == y else x < y).

Lemma ltr_asym x y : x < y < x = false.

Lemma ler_anti : antisymmetric (@ler R).

Lemma ltr_le_asym x y : x < y x = false.

Lemma ler_lt_asym x y : x y < x = false.

Definition lter_anti := (=^~ eqr_le, ltr_asym, ltr_le_asym, ler_lt_asym).

Lemma ltr_geF x y : x < y (y x = false).

Lemma ler_gtF x y : x y (y < x = false).

Definition ltr_gtF x y hxy := ler_gtF (@ltrW x y hxy).

Norm and order properties.

Lemma normr_le0 x : (`|x| 0) = (x == 0).

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

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

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

End NumIntegralDomainTheory.

Hint Resolve @ler01 @ltr01 lerr ltrr ltrW ltr_eqF ltr0Sn ler0n normr_ge0 : core.

Section NumIntegralDomainMonotonyTheory.

Variables R R' : numDomainType.
Implicit Types m n p : nat.
Implicit Types x y z : R.
Implicit Types u v w : R'.

This listing of "Let"s factor out the required premices for the subsequent lemmas, putting them in the context so that "done" solves the goals quickly

Let leqnn := leqnn.
Let ltnE := ltn_neqAle.
Let ltrE := @ltr_neqAle R.
Let ltr'E := @ltr_neqAle R'.
Let gtnE (m n : nat) : (m > n)%N = (m != n) && (m n)%N.
Let gtrE (x y : R) : (x > y) = (x != y) && (x y).
Let gtr'E (x y : R') : (x > y) = (x != y) && (x y).
Let leq_anti : antisymmetric leq.
Let geq_anti : antisymmetric geq.
Let ler_antiR := @ler_anti R.
Let ler_antiR' := @ler_anti R'.
Let ger_antiR : antisymmetric (>=%R : rel R).
Let ger_antiR' : antisymmetric (>=%R : rel R').
Let leq_total := leq_total.
Let geq_total : total geq.

Section AcrossTypes.

Variables (D D' : {pred R}) (f : R R').

Lemma ltrW_homo : {homo f : x y / x < y} {homo f : x y / x y}.

Lemma ltrW_nhomo : {homo f : x y /~ x < y} {homo f : x y /~ x y}.

Lemma inj_homo_ltr :
  injective f {homo f : x y / x y} {homo f : x y / x < y}.

Lemma inj_nhomo_ltr :
  injective f {homo f : x y /~ x y} {homo f : x y /~ x < y}.

Lemma incr_inj : {mono f : x y / x y} injective f.

Lemma decr_inj : {mono f : x y /~ x y} injective f.

Lemma lerW_mono : {mono f : x y / x y} {mono f : x y / x < y}.

Lemma lerW_nmono : {mono f : x y /~ x y} {mono f : x y /~ x < y}.

Monotony in D D'
Lemma ltrW_homo_in :
  {in D & D', {homo f : x y / x < y}} {in D & D', {homo f : x y / x y}}.

Lemma ltrW_nhomo_in :
  {in D & D', {homo f : x y /~ x < y}} {in D & D', {homo f : x y /~ x y}}.

Lemma inj_homo_ltr_in :
    {in D & D', injective f} {in D & D', {homo f : x y / x y}}
  {in D & D', {homo f : x y / x < y}}.

Lemma inj_nhomo_ltr_in :
    {in D & D', injective f} {in D & D', {homo f : x y /~ x y}}
  {in D & D', {homo f : x y /~ x < y}}.

Lemma incr_inj_in : {in D &, {mono f : x y / x y}}
   {in D &, injective f}.

Lemma decr_inj_in :
  {in D &, {mono f : x y /~ x y}} {in D &, injective f}.

Lemma lerW_mono_in :
  {in D &, {mono f : x y / x y}} {in D &, {mono f : x y / x < y}}.

Lemma lerW_nmono_in :
  {in D &, {mono f : x y /~ x y}} {in D &, {mono f : x y /~ x < y}}.

End AcrossTypes.

Section NatToR.

Variables (D D' : {pred nat}) (f : nat R).

Lemma ltnrW_homo : {homo f : m n / (m < n)%N >-> m < n}
  {homo f : m n / (m n)%N >-> m n}.

Lemma ltnrW_nhomo : {homo f : m n / (n < m)%N >-> m < n}
  {homo f : m n / (n m)%N >-> m n}.

Lemma inj_homo_ltnr : injective f
  {homo f : m n / (m n)%N >-> m n}
  {homo f : m n / (m < n)%N >-> m < n}.

Lemma inj_nhomo_ltnr : injective f
  {homo f : m n / (n m)%N >-> m n}
  {homo f : m n / (n < m)%N >-> m < n}.

Lemma incnr_inj : {mono f : m n / (m n)%N >-> m n} injective f.

Lemma decnr_inj_inj : {mono f : m n / (n m)%N >-> m n} injective f.

Lemma lenrW_mono : {mono f : m n / (m n)%N >-> m n}
  {mono f : m n / (m < n)%N >-> m < n}.

Lemma lenrW_nmono : {mono f : m n / (n m)%N >-> m n}
  {mono f : m n / (n < m)%N >-> m < n}.

Lemma lenr_mono : {homo f : m n / (m < n)%N >-> m < n}
   {mono f : m n / (m n)%N >-> m n}.

Lemma lenr_nmono : {homo f : m n / (n < m)%N >-> m < n}
  {mono f : m n / (n m)%N >-> m n}.

Lemma ltnrW_homo_in : {in D & D', {homo f : m n / (m < n)%N >-> m < n}}
  {in D & D', {homo f : m n / (m n)%N >-> m n}}.

Lemma ltnrW_nhomo_in : {in D & D', {homo f : m n / (n < m)%N >-> m < n}}
  {in D & D', {homo f : m n / (n m)%N >-> m n}}.

Lemma inj_homo_ltnr_in : {in D & D', injective f}
  {in D & D', {homo f : m n / (m n)%N >-> m n}}
  {in D & D', {homo f : m n / (m < n)%N >-> m < n}}.

Lemma inj_nhomo_ltnr_in : {in D & D', injective f}
  {in D & D', {homo f : m n / (n m)%N >-> m n}}
  {in D & D', {homo f : m n / (n < m)%N >-> m < n}}.

Lemma incnr_inj_in : {in D &, {mono f : m n / (m n)%N >-> m n}}
  {in D &, injective f}.

Lemma decnr_inj_inj_in : {in D &, {mono f : m n / (n m)%N >-> m n}}
  {in D &, injective f}.

Lemma lenrW_mono_in : {in D &, {mono f : m n / (m n)%N >-> m n}}
  {in D &, {mono f : m n / (m < n)%N >-> m < n}}.

Lemma lenrW_nmono_in : {in D &, {mono f : m n / (n m)%N >-> m n}}
  {in D &, {mono f : m n / (n < m)%N >-> m < n}}.

Lemma lenr_mono_in : {in D &, {homo f : m n / (m < n)%N >-> m < n}}
   {in D &, {mono f : m n / (m n)%N >-> m n}}.

Lemma lenr_nmono_in : {in D &, {homo f : m n / (n < m)%N >-> m < n}}
  {in D &, {mono f : m n / (n m)%N >-> m n}}.

End NatToR.

Section RToNat.

Variables (D D' : {pred R}) (f : R nat).

Lemma ltrnW_homo : {homo f : m n / m < n >-> (m < n)%N}
  {homo f : m n / m n >-> (m n)%N}.

Lemma ltrnW_nhomo : {homo f : m n / n < m >-> (m < n)%N}
  {homo f : m n / n m >-> (m n)%N}.

Lemma inj_homo_ltrn : injective f
  {homo f : m n / m n >-> (m n)%N}
  {homo f : m n / m < n >-> (m < n)%N}.

Lemma inj_nhomo_ltrn : injective f
  {homo f : m n / n m >-> (m n)%N}
  {homo f : m n / n < m >-> (m < n)%N}.

Lemma incrn_inj : {mono f : m n / m n >-> (m n)%N} injective f.

Lemma decrn_inj : {mono f : m n / n m >-> (m n)%N} injective f.

Lemma lernW_mono : {mono f : m n / m n >-> (m n)%N}
  {mono f : m n / m < n >-> (m < n)%N}.

Lemma lernW_nmono : {mono f : m n / n m >-> (m n)%N}
  {mono f : m n / n < m >-> (m < n)%N}.

Lemma ltrnW_homo_in : {in D & D', {homo f : m n / m < n >-> (m < n)%N}}
  {in D & D', {homo f : m n / m n >-> (m n)%N}}.

Lemma ltrnW_nhomo_in : {in D & D', {homo f : m n / n < m >-> (m < n)%N}}
  {in D & D', {homo f : m n / n m >-> (m n)%N}}.

Lemma inj_homo_ltrn_in : {in D & D', injective f}
  {in D & D', {homo f : m n / m n >-> (m n)%N}}
  {in D & D', {homo f : m n / m < n >-> (m < n)%N}}.

Lemma inj_nhomo_ltrn_in : {in D & D', injective f}
  {in D & D', {homo f : m n / n m >-> (m n)%N}}
  {in D & D', {homo f : m n / n < m >-> (m < n)%N}}.

Lemma incrn_inj_in : {in D &, {mono f : m n / m n >-> (m n)%N}}
  {in D &, injective f}.

Lemma decrn_inj_in : {in D &, {mono f : m n / n m >-> (m n)%N}}
  {in D &, injective f}.

Lemma lernW_mono_in : {in D &, {mono f : m n / m n >-> (m n)%N}}
  {in D &, {mono f : m n / m < n >-> (m < n)%N}}.

Lemma lernW_nmono_in : {in D &, {mono f : m n / n m >-> (m n)%N}}
  {in D &, {mono f : m n / n < m >-> (m < n)%N}}.

End RToNat.

End NumIntegralDomainMonotonyTheory.

Section NumDomainOperationTheory.

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

Comparision and opposite.

Lemma ler_opp2 : {mono -%R : x y /~ x y :> R}.
Hint Resolve ler_opp2 : core.
Lemma ltr_opp2 : {mono -%R : x y /~ x < y :> R}.
Hint Resolve ltr_opp2 : core.
Definition lter_opp2 := (ler_opp2, ltr_opp2).

Lemma ler_oppr x y : (x - y) = (y - x).

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

Definition lter_oppr := (ler_oppr, ltr_oppr).

Lemma ler_oppl x y : (- x y) = (- y x).

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

Definition lter_oppl := (ler_oppl, ltr_oppl).

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).

Definition oppr_lte0 := (oppr_le0, oppr_lt0).
Definition oppr_cp0 := (oppr_gte0, oppr_lte0).
Definition lter_oppE := (oppr_cp0, lter_opp2).

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.
Hint Resolve real0 : core.

Lemma real1 : 1 \is @real R.
Hint Resolve real1 : core.

Lemma realn n : n%:R \is @real R.

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

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

:TODO: add a rpredBC in ssralg
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 bool bool Set :=
  | LerNotGt of x y : ler_xor_gt x y (y - x) (y - x) true false
  | GtrNotLe of y < x : ler_xor_gt x y (x - y) (x - y) false true.

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

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

Lemma real_lerP x y :
    x \is real y \is real
  ler_xor_gt x y `|x - y| `|y - x| (x y) (y < x).

Lemma real_ltrP x y :
    x \is real y \is real
  ltr_xor_ge x y `|x - y| `|y - x| (y x) (x < y).

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

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

Lemma real_ltrgtP x y :
    x \is real y \is real
  comparer x y `|x - y| `|y - x|
                (y == x) (x == y) (x y) (y x) (x < y) (x > y).

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

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

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

Lemma real_ger0P x : x \is real ger0_xor_lt0 x `|x| (x < 0) (0 x).

Lemma real_ler0P x : x \is real ler0_xor_gt0 x `|x| (0 < x) (x 0).

Lemma real_ltrgt0P x :
     x \is real
  comparer0 x `|x| (0 == x) (x == 0) (x 0) (0 x) (x < 0) (x > 0).

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

Lemma ler_sub_real x y : x y y - x \is real.

Lemma ger_sub_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 ler_add2l x : {mono +%R x : y z / y z}.

Lemma ler_add2r x : {mono +%R^~ x : y z / y z}.

Lemma ltr_add2l x : {mono +%R x : y z / y < z}.

Lemma ltr_add2r x : {mono +%R^~ x : y z / y < z}.

Definition ler_add2 := (ler_add2l, ler_add2r).
Definition ltr_add2 := (ltr_add2l, ltr_add2r).
Definition lter_add2 := (ler_add2, ltr_add2).

Addition, subtraction and transitivity
Lemma ler_add x y z t : x y z t x + z y + t.

Lemma ler_lt_add x y z t : x y z < t x + z < y + t.

Lemma ltr_le_add x y z t : x < y z t x + z < y + t.

Lemma ltr_add x y z t : x < y z < t x + z < y + t.

Lemma ler_sub x y z t : x y t z x - z y - t.

Lemma ler_lt_sub x y z t : x y t < z x - z < y - t.

Lemma ltr_le_sub x y z t : x < y t z x - z < y - t.

Lemma ltr_sub x y z t : x < y t < z x - z < y - t.

Lemma ler_subl_addr x y z : (x - y z) = (x z + y).

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

Lemma ler_subr_addr x y z : (x y - z) = (x + z y).

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

Definition ler_sub_addr := (ler_subl_addr, ler_subr_addr).
Definition ltr_sub_addr := (ltr_subl_addr, ltr_subr_addr).
Definition lter_sub_addr := (ler_sub_addr, ltr_sub_addr).

Lemma ler_subl_addl x y z : (x - y z) = (x y + z).

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

Lemma ler_subr_addl x y z : (x y - z) = (z + x y).

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

Definition ler_sub_addl := (ler_subl_addl, ler_subr_addl).
Definition ltr_sub_addl := (ltr_subl_addl, ltr_subr_addl).
Definition lter_sub_addl := (ler_sub_addl, ltr_sub_addl).

Lemma ler_addl x y : (x x + y) = (0 y).

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

Lemma ler_addr x y : (x y + x) = (0 y).

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

Lemma ger_addl x y : (x + y x) = (y 0).

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

Lemma ger_addr x y : (y + x x) = (y 0).

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

Definition cpr_add := (ler_addl, ler_addr, ger_addl, ger_addl,
                       ltr_addl, ltr_addr, gtr_addl, gtr_addl).

Addition with left member knwon to be positive/negative
Lemma ler_paddl y x z : 0 x y z y x + z.

Lemma ltr_paddl y x z : 0 x y < z y < x + z.

Lemma ltr_spaddl y x z : 0 < x y z y < x + z.

Lemma ltr_spsaddl y x z : 0 < x y < z y < x + z.

Lemma ler_naddl y x z : x 0 y z x + y z.

Lemma ltr_naddl y x z : x 0 y < z x + y < z.

Lemma ltr_snaddl y x z : x < 0 y z x + y < z.

Lemma ltr_snsaddl y x z : x < 0 y < z x + y < z.

Addition with right member we know positive/negative
Lemma ler_paddr y x z : 0 x y z y z + x.

Lemma ltr_paddr y x z : 0 x y < z y < z + x.

Lemma ltr_spaddr y x z : 0 < x y z y < z + x.

Lemma ltr_spsaddr y x z : 0 < x y < z y < z + x.

Lemma ler_naddr y x z : x 0 y z y + x z.

Lemma ltr_naddr y x z : x 0 y < z y + x < z.

Lemma ltr_snaddr y x z : x < 0 y z y + x < z.

Lemma ltr_snsaddr 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