Library mathcomp.algebra.ssrnum
+ +
+(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
+ Distributed under the terms of CeCILL-B. *)
+Require Import mathcomp.ssreflect.ssreflect.
+ +
+
+
++ Distributed under the terms of CeCILL-B. *)
+Require Import mathcomp.ssreflect.ssreflect.
+ +
+
+
+
+
+ This file defines some classes to manipulate number structures, i.e
+ structures with an order and a norm
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 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 (<).
+
+
+
+
+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. + +-
+
- 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 + + +
+
+
+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.
+ +
+
+
++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)
+}.
+ +
+ +
+
+
++ 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; _ : Type}.
+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 T.
+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 m ⇒ Pack (@Class T b m) T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+ +
+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.
+ +
+
+
++ +
+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; _ : Type}.
+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 T.
+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 m ⇒ Pack (@Class T b m) T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+ +
+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.
+ +
+
+
++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.
+ +
+ +
+
+
++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; _ : Type}.
+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) T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+Definition numDomainType := @NumDomain.Pack cT xclass xT.
+Definition fieldType := @GRing.Field.Pack cT xclass xT.
+Definition join_numDomainType := @NumDomain.Pack fieldType xclass xT.
+ +
+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.
+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; _ : Type}.
+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 mc ⇒ Pack (@Class T b m mc) T.
+Definition clone := fun b & phant_id class (b : class_of T) ⇒ Pack b T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+Definition numDomainType := @NumDomain.Pack cT xclass xT.
+Definition fieldType := @GRing.Field.Pack cT xclass xT.
+Definition numFieldType := @NumField.Pack cT xclass xT.
+Definition decFieldType := @GRing.DecidableField.Pack cT xclass xT.
+Definition closedFieldType := @GRing.ClosedField.Pack cT xclass xT.
+Definition join_dec_numDomainType := @NumDomain.Pack decFieldType xclass xT.
+Definition join_dec_numFieldType := @NumField.Pack decFieldType xclass xT.
+Definition join_numDomainType := @NumDomain.Pack closedFieldType xclass xT.
+Definition join_numFieldType := @NumField.Pack closedFieldType xclass xT.
+ +
+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; _ : Type}.
+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 T.
+Definition pack b0 (m0 : real_axiom (num_for T b0)) :=
+ fun bT b & phant_id (NumDomain.class bT) b ⇒
+ fun m & phant_id m0 m ⇒ Pack (@Class T b m) T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+Definition numDomainType := @NumDomain.Pack cT xclass xT.
+ +
+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; _ : Type}.
+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) T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+Definition numDomainType := @NumDomain.Pack cT xclass xT.
+Definition realDomainType := @RealDomain.Pack cT xclass xT.
+Definition fieldType := @GRing.Field.Pack cT xclass xT.
+Definition numFieldType := @NumField.Pack cT xclass xT.
+Definition join_realDomainType := @RealDomain.Pack numFieldType xclass xT.
+ +
+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_realDomainType.
+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; _ : Type}.
+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 T.
+Definition pack b0 (m0 : archimedean_axiom (num_for T b0)) :=
+ fun bT b & phant_id (RealField.class bT) b ⇒
+ fun m & phant_id m0 m ⇒ Pack (@Class T b m) T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+Definition numDomainType := @NumDomain.Pack cT xclass xT.
+Definition realDomainType := @RealDomain.Pack cT xclass xT.
+Definition fieldType := @GRing.Field.Pack cT xclass xT.
+Definition numFieldType := @NumField.Pack cT xclass xT.
+Definition realFieldType := @RealField.Pack cT xclass xT.
+ +
+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; _ : Type}.
+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 T.
+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 m ⇒ Pack (@Class T b m) T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+Definition numDomainType := @NumDomain.Pack cT xclass xT.
+Definition realDomainType := @RealDomain.Pack cT xclass xT.
+Definition fieldType := @GRing.Field.Pack cT xclass xT.
+Definition numFieldType := @NumField.Pack cT xclass xT.
+Definition realFieldType := @RealField.Pack cT xclass xT.
+ +
+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.
+ +
+
+
++ +
+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; _ : Type}.
+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) T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+Definition numDomainType := @NumDomain.Pack cT xclass xT.
+Definition fieldType := @GRing.Field.Pack cT xclass xT.
+Definition join_numDomainType := @NumDomain.Pack fieldType xclass xT.
+ +
+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.
+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; _ : Type}.
+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 mc ⇒ Pack (@Class T b m mc) T.
+Definition clone := fun b & phant_id class (b : class_of T) ⇒ Pack b T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+Definition numDomainType := @NumDomain.Pack cT xclass xT.
+Definition fieldType := @GRing.Field.Pack cT xclass xT.
+Definition numFieldType := @NumField.Pack cT xclass xT.
+Definition decFieldType := @GRing.DecidableField.Pack cT xclass xT.
+Definition closedFieldType := @GRing.ClosedField.Pack cT xclass xT.
+Definition join_dec_numDomainType := @NumDomain.Pack decFieldType xclass xT.
+Definition join_dec_numFieldType := @NumField.Pack decFieldType xclass xT.
+Definition join_numDomainType := @NumDomain.Pack closedFieldType xclass xT.
+Definition join_numFieldType := @NumField.Pack closedFieldType xclass xT.
+ +
+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; _ : Type}.
+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 T.
+Definition pack b0 (m0 : real_axiom (num_for T b0)) :=
+ fun bT b & phant_id (NumDomain.class bT) b ⇒
+ fun m & phant_id m0 m ⇒ Pack (@Class T b m) T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+Definition numDomainType := @NumDomain.Pack cT xclass xT.
+ +
+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; _ : Type}.
+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) T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+Definition numDomainType := @NumDomain.Pack cT xclass xT.
+Definition realDomainType := @RealDomain.Pack cT xclass xT.
+Definition fieldType := @GRing.Field.Pack cT xclass xT.
+Definition numFieldType := @NumField.Pack cT xclass xT.
+Definition join_realDomainType := @RealDomain.Pack numFieldType xclass xT.
+ +
+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_realDomainType.
+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; _ : Type}.
+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 T.
+Definition pack b0 (m0 : archimedean_axiom (num_for T b0)) :=
+ fun bT b & phant_id (RealField.class bT) b ⇒
+ fun m & phant_id m0 m ⇒ Pack (@Class T b m) T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+Definition numDomainType := @NumDomain.Pack cT xclass xT.
+Definition realDomainType := @RealDomain.Pack cT xclass xT.
+Definition fieldType := @GRing.Field.Pack cT xclass xT.
+Definition numFieldType := @NumField.Pack cT xclass xT.
+Definition realFieldType := @RealField.Pack cT xclass xT.
+ +
+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; _ : Type}.
+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 T.
+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 m ⇒ Pack (@Class T b m) T.
+ +
+Definition eqType := @Equality.Pack cT xclass xT.
+Definition choiceType := @Choice.Pack cT xclass xT.
+Definition zmodType := @GRing.Zmodule.Pack cT xclass xT.
+Definition ringType := @GRing.Ring.Pack cT xclass xT.
+Definition comRingType := @GRing.ComRing.Pack cT xclass xT.
+Definition unitRingType := @GRing.UnitRing.Pack cT xclass xT.
+Definition comUnitRingType := @GRing.ComUnitRing.Pack cT xclass xT.
+Definition idomainType := @GRing.IntegralDomain.Pack cT xclass xT.
+Definition numDomainType := @NumDomain.Pack cT xclass xT.
+Definition realDomainType := @RealDomain.Pack cT xclass xT.
+Definition fieldType := @GRing.Field.Pack cT xclass xT.
+Definition numFieldType := @NumField.Pack cT xclass xT.
+Definition realFieldType := @RealField.Pack cT xclass xT.
+ +
+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.
+ +
+
+
++ +
+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).
+ +
+
+
++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).
+ +
+
+
++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 return bool then y ^+ 2 == x else y == 0.
+ +
+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.
+ +
+
+
++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 return bool then y ^+ 2 == x else y == 0.
+ +
+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.
+ +
+
+
++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).
+ +
+
+
++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.
+ +
+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).
+ +
+
+
++Lemma lerr x : x ≤ x.
+Lemma ltrr x : x < x = false.
+Lemma ltrW x y : x < y → x ≤ y.
+Hint Resolve lerr ltrr ltrW.
+ +
+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.
+Lemma ltr0Sn n : 0 < n.+1%:R :> R.
+ Lemma ltr0n n : (0 < n%:R :> R) = (0 < n)%N.
+ Hint Resolve ltr0Sn.
+ +
+Lemma pnatr_eq0 n : (n%:R == 0 :> R) = (n == 0)%N.
+ +
+Lemma char_num : [char R] =i pred0.
+ +
+
+
++Lemma ltr01 : 0 < 1 :> R.
+Lemma ler0n n : 0 ≤ n%:R :> R.
+Hint Resolve ler01 ltr01 ler0n.
+Lemma ltr0Sn n : 0 < n.+1%:R :> R.
+ Lemma ltr0n n : (0 < n%:R :> R) = (0 < n)%N.
+ Hint Resolve ltr0Sn.
+ +
+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.
+ +
+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).
+ +
+
+
++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.
+ +
+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).
+ +
+
+
++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).
+ +
+
+
++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.
+ +
+Section NumIntegralDomainMonotonyTheory.
+ +
+Variables R R' : numDomainType.
+Implicit Types m n p : nat.
+Implicit Types x y z : R.
+Implicit Types u v w : R'.
+ +
+Section AcrossTypes.
+ +
+Variable D D' : pred R.
+Variable (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 homo_inj_lt :
+ injective f → {homo f : x y / x ≤ y} → {homo f : x y / x < y}.
+ +
+Lemma nhomo_inj_lt :
+ injective f → {homo f : x y /~ x ≤ y} → {homo f : x y /~ x < y}.
+ +
+Lemma mono_inj : {mono f : x y / x ≤ y} → injective f.
+ +
+Lemma nmono_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}.
+ +
+
+
++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.
+ +
+Section NumIntegralDomainMonotonyTheory.
+ +
+Variables R R' : numDomainType.
+Implicit Types m n p : nat.
+Implicit Types x y z : R.
+Implicit Types u v w : R'.
+ +
+Section AcrossTypes.
+ +
+Variable D D' : pred R.
+Variable (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 homo_inj_lt :
+ injective f → {homo f : x y / x ≤ y} → {homo f : x y / x < y}.
+ +
+Lemma nhomo_inj_lt :
+ injective f → {homo f : x y /~ x ≤ y} → {homo f : x y /~ x < y}.
+ +
+Lemma mono_inj : {mono f : x y / x ≤ y} → injective f.
+ +
+Lemma nmono_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 homo_inj_in_lt :
+ {in D & D', injective f} → {in D & D', {homo f : x y / x ≤ y}} →
+ {in D & D', {homo f : x y / x < y}}.
+ +
+Lemma nhomo_inj_in_lt :
+ {in D & D', injective f} → {in D & D', {homo f : x y /~ x ≤ y}} →
+ {in D & D', {homo f : x y /~ x < y}}.
+ +
+Lemma mono_inj_in : {in D &, {mono f : x y / x ≤ y}} → {in D &, injective f}.
+ +
+Lemma nmono_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.
+ +
+Variable (f : nat → R).
+ +
+Lemma ltn_ltrW_homo :
+ {homo f : m n / (m < n)%N >-> m < n} →
+ {homo f : m n / (m ≤ n)%N >-> m ≤ n}.
+ +
+Lemma ltn_ltrW_nhomo :
+ {homo f : m n / (n < m)%N >-> m < n} →
+ {homo f : m n / (n ≤ m)%N >-> m ≤ n}.
+ +
+Lemma homo_inj_ltn_lt :
+ injective f → {homo f : m n / (m ≤ n)%N >-> m ≤ n} →
+ {homo f : m n / (m < n)%N >-> m < n}.
+ +
+Lemma nhomo_inj_ltn_lt :
+ injective f → {homo f : m n / (n ≤ m)%N >-> m ≤ n} →
+ {homo f : m n / (n < m)%N >-> m < n}.
+ +
+Lemma leq_mono_inj : {mono f : m n / (m ≤ n)%N >-> m ≤ n} → injective f.
+ +
+Lemma leq_nmono_inj : {mono f : m n / (n ≤ m)%N >-> m ≤ n} → injective f.
+ +
+Lemma leq_lerW_mono :
+ {mono f : m n / (m ≤ n)%N >-> m ≤ n} →
+ {mono f : m n / (m < n)%N >-> m < n}.
+ +
+Lemma leq_lerW_nmono :
+ {mono f : m n / (n ≤ m)%N >-> m ≤ n} →
+ {mono f : m n / (n < m)%N >-> m < n}.
+ +
+Lemma homo_leq_mono :
+ {homo f : m n / (m < n)%N >-> m < n} →
+ {mono f : m n / (m ≤ n)%N >-> m ≤ n}.
+ +
+Lemma nhomo_leq_mono :
+ {homo f : m n / (n < m)%N >-> m < n} →
+ {mono f : m n / (n ≤ m)%N >-> m ≤ n}.
+ +
+End NatToR.
+ +
+End NumIntegralDomainMonotonyTheory.
+ +
+Section NumDomainOperationTheory.
+ +
+Variable R : numDomainType.
+Implicit Types x y z t : R.
+ +
+
+
++ {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 homo_inj_in_lt :
+ {in D & D', injective f} → {in D & D', {homo f : x y / x ≤ y}} →
+ {in D & D', {homo f : x y / x < y}}.
+ +
+Lemma nhomo_inj_in_lt :
+ {in D & D', injective f} → {in D & D', {homo f : x y /~ x ≤ y}} →
+ {in D & D', {homo f : x y /~ x < y}}.
+ +
+Lemma mono_inj_in : {in D &, {mono f : x y / x ≤ y}} → {in D &, injective f}.
+ +
+Lemma nmono_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.
+ +
+Variable (f : nat → R).
+ +
+Lemma ltn_ltrW_homo :
+ {homo f : m n / (m < n)%N >-> m < n} →
+ {homo f : m n / (m ≤ n)%N >-> m ≤ n}.
+ +
+Lemma ltn_ltrW_nhomo :
+ {homo f : m n / (n < m)%N >-> m < n} →
+ {homo f : m n / (n ≤ m)%N >-> m ≤ n}.
+ +
+Lemma homo_inj_ltn_lt :
+ injective f → {homo f : m n / (m ≤ n)%N >-> m ≤ n} →
+ {homo f : m n / (m < n)%N >-> m < n}.
+ +
+Lemma nhomo_inj_ltn_lt :
+ injective f → {homo f : m n / (n ≤ m)%N >-> m ≤ n} →
+ {homo f : m n / (n < m)%N >-> m < n}.
+ +
+Lemma leq_mono_inj : {mono f : m n / (m ≤ n)%N >-> m ≤ n} → injective f.
+ +
+Lemma leq_nmono_inj : {mono f : m n / (n ≤ m)%N >-> m ≤ n} → injective f.
+ +
+Lemma leq_lerW_mono :
+ {mono f : m n / (m ≤ n)%N >-> m ≤ n} →
+ {mono f : m n / (m < n)%N >-> m < n}.
+ +
+Lemma leq_lerW_nmono :
+ {mono f : m n / (n ≤ m)%N >-> m ≤ n} →
+ {mono f : m n / (n < m)%N >-> m < n}.
+ +
+Lemma homo_leq_mono :
+ {homo f : m n / (m < n)%N >-> m < n} →
+ {mono f : m n / (m ≤ n)%N >-> m ≤ n}.
+ +
+Lemma nhomo_leq_mono :
+ {homo f : m n / (n < m)%N >-> m < n} →
+ {mono f : m n / (n ≤ m)%N >-> m ≤ n}.
+ +
+End NatToR.
+ +
+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.
+Lemma ltr_opp2 : {mono -%R : x y /~ x < y :> R}.
+ Hint Resolve ltr_opp2.
+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).
+ +
+
+
++Lemma ler_opp2 : {mono -%R : x y /~ x ≤ y :> R}.
+ Hint Resolve ler_opp2.
+Lemma ltr_opp2 : {mono -%R : x y /~ x < y :> R}.
+ Hint Resolve ltr_opp2.
+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.
+ +
+Lemma real1 : 1 \is @real R.
+Hint Resolve real1.
+ +
+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}.
+ +
+
+
++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.
+ +
+Lemma real1 : 1 \is @real R.
+Hint Resolve real1.
+ +
+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}.
+ +
+
+
++ +
+Lemma realD : {in real &, ∀ x y, x + y \is real}.
+ +
+
+ dichotomy and trichotomy
+
+
+
+
+CoInductive 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.
+ +
+CoInductive 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.
+ +
+CoInductive 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).
+ +
+CoInductive 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.
+ +
+CoInductive 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.
+ +
+CoInductive 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.
+ +
+
+
++CoInductive 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.
+ +
+CoInductive 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.
+ +
+CoInductive 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).
+ +
+CoInductive 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.
+ +
+CoInductive 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.
+ +
+CoInductive 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.
+ +
+
+
++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_add2r z x y : (x + z < y + z) = (x < y).
+ +
+Lemma ltr_add2l z x y : (z + x < z + y) = (x < y).
+ +
+Definition ler_add2 := (ler_add2l, ler_add2r).
+Definition ltr_add2 := (ltr_add2l, ltr_add2r).
+Definition lter_add2 := (ler_add2, ltr_add2).
+ +
+
+
++ +
+Lemma ler_add2r x : {mono +%R^~ x : y z / y ≤ z}.
+ +
+Lemma ltr_add2r z x y : (x + z < y + z) = (x < y).
+ +
+Lemma ltr_add2l z x y : (z + x < z + y) = (x < y).
+ +
+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).
+ +
+
+
++ +
+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.
+ +
+
+
++ +
+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.
+ +
+
+
++ +
+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).
+ +
+
+
++ 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 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).
+ +
+
+
++ (∀ 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 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).
+ +
+
+
++ (∀ i, P i → 0 ≤ F i) → \sum_(i | P i) F i = 0 →
+ (∀ i, P i → F i = 0).
+ +
+
+ mulr and ler/ltr
+
+
+
+
+Lemma ler_pmul2l x : 0 < x → {mono *%R x : x y / x ≤ y}.
+ +
+Lemma ltr_pmul2l x : 0 < x → {mono *%R x : x y / x < y}.
+ +
+Definition lter_pmul2l := (ler_pmul2l, ltr_pmul2l).
+ +
+Lemma ler_pmul2r x : 0 < x → {mono *%R^~ x : x y / x ≤ y}.
+ +
+Lemma ltr_pmul2r x : 0 < x → {mono *%R^~ x : x y / x < y}.
+ +
+Definition lter_pmul2r := (ler_pmul2r, ltr_pmul2r).
+ +
+Lemma ler_nmul2l x : x < 0 → {mono *%R x : x y /~ x ≤ y}.
+ +
+Lemma ltr_nmul2l x : x < 0 → {mono *%R x : x y /~ x < y}.
+ +
+Definition lter_nmul2l := (ler_nmul2l, ltr_nmul2l).
+ +
+Lemma ler_nmul2r x : x < 0 → {mono *%R^~ x : x y /~ x ≤ y}.
+ +
+Lemma ltr_nmul2r x : x < 0 → {mono *%R^~ x : x y /~ x < y}.
+ +
+Definition lter_nmul2r := (ler_nmul2r, ltr_nmul2r).
+ +
+Lemma ler_wpmul2l x : 0 ≤ x → {homo *%R x : y z / y ≤ z}.
+ +
+Lemma ler_wpmul2r x : 0 ≤ x → {homo *%R^~ x : y z / y ≤ z}.
+ +
+Lemma ler_wnmul2l x : x ≤ 0 → {homo *%R x : y z /~ y ≤ z}.
+ +
+Lemma ler_wnmul2r x : x ≤ 0 → {homo *%R^~ x : y z /~ y ≤ z}.
+ +
+
+
++Lemma ler_pmul2l x : 0 < x → {mono *%R x : x y / x ≤ y}.
+ +
+Lemma ltr_pmul2l x : 0 < x → {mono *%R x : x y / x < y}.
+ +
+Definition lter_pmul2l := (ler_pmul2l, ltr_pmul2l).
+ +
+Lemma ler_pmul2r x : 0 < x → {mono *%R^~ x : x y / x ≤ y}.
+ +
+Lemma ltr_pmul2r x : 0 < x → {mono *%R^~ x : x y / x < y}.
+ +
+Definition lter_pmul2r := (ler_pmul2r, ltr_pmul2r).
+ +
+Lemma ler_nmul2l x : x < 0 → {mono *%R x : x y /~ x ≤ y}.
+ +
+Lemma ltr_nmul2l x : x < 0 → {mono *%R x : x y /~ x < y}.
+ +
+Definition lter_nmul2l := (ler_nmul2l, ltr_nmul2l).
+ +
+Lemma ler_nmul2r x : x < 0 → {mono *%R^~ x : x y /~ x ≤ y}.
+ +
+Lemma ltr_nmul2r x : x < 0 → {mono *%R^~ x : x y /~ x < y}.
+ +
+Definition lter_nmul2r := (ler_nmul2r, ltr_nmul2r).
+ +
+Lemma ler_wpmul2l x : 0 ≤ x → {homo *%R x : y z / y ≤ z}.
+ +
+Lemma ler_wpmul2r x : 0 ≤ x → {homo *%R^~ x : y z / y ≤ z}.
+ +
+Lemma ler_wnmul2l x : x ≤ 0 → {homo *%R x : y z /~ y ≤ z}.
+ +
+Lemma ler_wnmul2r x : x ≤ 0 → {homo *%R^~ x : y z /~ y ≤ z}.
+ +
+
+ Binary forms, for backchaining.
+
+
+
+
+Lemma ler_pmul x1 y1 x2 y2 :
+ 0 ≤ x1 → 0 ≤ x2 → x1 ≤ y1 → x2 ≤ y2 → x1 × x2 ≤ y1 × y2.
+ +
+Lemma ltr_pmul x1 y1 x2 y2 :
+ 0 ≤ x1 → 0 ≤ x2 → x1 < y1 → x2 < y2 → x1 × x2 < y1 × y2.
+ +
+
+
++Lemma ler_pmul x1 y1 x2 y2 :
+ 0 ≤ x1 → 0 ≤ x2 → x1 ≤ y1 → x2 ≤ y2 → x1 × x2 ≤ y1 × y2.
+ +
+Lemma ltr_pmul x1 y1 x2 y2 :
+ 0 ≤ x1 → 0 ≤ x2 → x1 < y1 → x2 < y2 → x1 × x2 < y1 × y2.
+ +
+
+ complement for x *+ n and <= or <
+
+
+
+
+Lemma ler_pmuln2r n : (0 < n)%N → {mono (@GRing.natmul R)^~ n : x y / x ≤ y}.
+ +
+Lemma ltr_pmuln2r 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_pmuln2r 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_wmuln2r x y n : x < y → (x *+ n < y *+ n) = (0 < n)%N.
+ +
+Lemma ltr_wpmuln2r n : (0 < n)%N → {homo (@GRing.natmul R)^~ n : x y / x < y}.
+ +
+Lemma ler_wmuln2r 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 ler_muln2r n x y : (x *+ n ≤ y *+ n) = ((n == 0%N) || (x ≤ y)).
+ +
+Lemma ltr_muln2r n x y : (x *+ n < y *+ n) = ((0 < n)%N && (x < y)).
+ +
+Lemma eqr_muln2r n x y : (x *+ n == y *+ n) = (n == 0)%N || (x == y).
+ +
+
+
++Lemma ler_pmuln2r n : (0 < n)%N → {mono (@GRing.natmul R)^~ n : x y / x ≤ y}.
+ +
+Lemma ltr_pmuln2r 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_pmuln2r 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_wmuln2r x y n : x < y → (x *+ n < y *+ n) = (0 < n)%N.
+ +
+Lemma ltr_wpmuln2r n : (0 < n)%N → {homo (@GRing.natmul R)^~ n : x y / x < y}.
+ +
+Lemma ler_wmuln2r 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 ler_muln2r n x y : (x *+ n ≤ y *+ n) = ((n == 0%N) || (x ≤ y)).
+ +
+Lemma ltr_muln2r n x y : (x *+ n < y *+ n) = ((0 < n)%N && (x < y)).
+ +
+Lemma eqr_muln2r 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 mulrIn x : x != 0 → injective (GRing.natmul x).
+ +
+Lemma ler_wpmuln2l x :
+ 0 ≤ x → {homo (@GRing.natmul R x) : m n / (m ≤ n)%N >-> m ≤ n}.
+ +
+Lemma ler_wnmuln2l 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_pmuln2l x :
+ 0 < x → {mono (@GRing.natmul R x) : m n / (m ≤ n)%N >-> m ≤ n}.
+ +
+Lemma ltr_pmuln2l x :
+ 0 < x → {mono (@GRing.natmul R x) : m n / (m < n)%N >-> m < n}.
+ +
+Lemma ler_nmuln2l x :
+ x < 0 → {mono (@GRing.natmul R x) : m n / (n ≤ m)%N >-> m ≤ n}.
+ +
+Lemma ltr_nmuln2l 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%N).
+ +
+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.
+ +
+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.
+ +
+
+
++Lemma mulrn_eq0 x n : (x *+ n == 0) = ((n == 0)%N || (x == 0)).
+ +
+Lemma mulrIn x : x != 0 → injective (GRing.natmul x).
+ +
+Lemma ler_wpmuln2l x :
+ 0 ≤ x → {homo (@GRing.natmul R x) : m n / (m ≤ n)%N >-> m ≤ n}.
+ +
+Lemma ler_wnmuln2l 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_pmuln2l x :
+ 0 < x → {mono (@GRing.natmul R x) : m n / (m ≤ n)%N >-> m ≤ n}.
+ +
+Lemma ltr_pmuln2l x :
+ 0 < x → {mono (@GRing.natmul R x) : m n / (m < n)%N >-> m < n}.
+ +
+Lemma ler_nmuln2l x :
+ x < 0 → {mono (@GRing.natmul R x) : m n / (n ≤ m)%N >-> m ≤ n}.
+ +
+Lemma ltr_nmuln2l 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%N).
+ +
+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.
+ +
+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).
+ +
+
+
++ +
+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).
+ +
+
+
++ +
+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).
+ +
+
+
++ +
+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).
+ +
+
+
++ +
+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.
+ +
+
+
++ +
+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
+
+
+
+
+ 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.
+ +
+
+
++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).
+ +
+
+
++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_pmull x y : 0 < y → (x × y ≤ y) = (x ≤ 1).
+ +
+Lemma gtr_pmull x y : 0 < y → (x × y < y) = (x < 1).
+ +
+Lemma ger_pmulr x y : 0 < y → (y × x ≤ y) = (x ≤ 1).
+ +
+Lemma gtr_pmulr x y : 0 < y → (y × x < y) = (x < 1).
+ +
+Lemma ler_pmull x y : 0 < y → (y ≤ x × y) = (1 ≤ x).
+ +
+Lemma ltr_pmull x y : 0 < y → (y < x × y) = (1 < x).
+ +
+Lemma ler_pmulr x y : 0 < y → (y ≤ y × x) = (1 ≤ x).
+ +
+Lemma ltr_pmulr x y : 0 < y → (y < y × x) = (1 < x).
+ +
+Lemma ger_nmull x y : y < 0 → (x × y ≤ y) = (1 ≤ x).
+ +
+Lemma gtr_nmull x y : y < 0 → (x × y < y) = (1 < x).
+ +
+Lemma ger_nmulr x y : y < 0 → (y × x ≤ y) = (1 ≤ x).
+ +
+Lemma gtr_nmulr x y : y < 0 → (y × x < y) = (1 < x).
+ +
+Lemma ler_nmull x y : y < 0 → (y ≤ x × y) = (x ≤ 1).
+ +
+Lemma ltr_nmull x y : y < 0 → (y < x × y) = (x < 1).
+ +
+Lemma ler_nmulr x y : y < 0 → (y ≤ y × x) = (x ≤ 1).
+ +
+Lemma ltr_nmulr x y : y < 0 → (y < y × x) = (x < 1).
+ +
+
+
++Lemma ger_pmull x y : 0 < y → (x × y ≤ y) = (x ≤ 1).
+ +
+Lemma gtr_pmull x y : 0 < y → (x × y < y) = (x < 1).
+ +
+Lemma ger_pmulr x y : 0 < y → (y × x ≤ y) = (x ≤ 1).
+ +
+Lemma gtr_pmulr x y : 0 < y → (y × x < y) = (x < 1).
+ +
+Lemma ler_pmull x y : 0 < y → (y ≤ x × y) = (1 ≤ x).
+ +
+Lemma ltr_pmull x y : 0 < y → (y < x × y) = (1 < x).
+ +
+Lemma ler_pmulr x y : 0 < y → (y ≤ y × x) = (1 ≤ x).
+ +
+Lemma ltr_pmulr x y : 0 < y → (y < y × x) = (1 < x).
+ +
+Lemma ger_nmull x y : y < 0 → (x × y ≤ y) = (1 ≤ x).
+ +
+Lemma gtr_nmull x y : y < 0 → (x × y < y) = (1 < x).
+ +
+Lemma ger_nmulr x y : y < 0 → (y × x ≤ y) = (1 ≤ x).
+ +
+Lemma gtr_nmulr x y : y < 0 → (y × x < y) = (1 < x).
+ +
+Lemma ler_nmull x y : y < 0 → (y ≤ x × y) = (x ≤ 1).
+ +
+Lemma ltr_nmull x y : y < 0 → (y < x × y) = (x < 1).
+ +
+Lemma ler_nmulr x y : y < 0 → (y ≤ y × x) = (x ≤ 1).
+ +
+Lemma ltr_nmulr 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_pemull x y : 0 ≤ y → 1 ≤ x → y ≤ x × y.
+ +
+Lemma ler_nemull x y : y ≤ 0 → 1 ≤ x → x × y ≤ y.
+ +
+Lemma ler_pemulr x y : 0 ≤ y → 1 ≤ x → y ≤ y × x.
+ +
+Lemma ler_nemulr x y : y ≤ 0 → 1 ≤ x → y × x ≤ y.
+ +
+Lemma ler_pimull x y : 0 ≤ y → x ≤ 1 → x × y ≤ y.
+ +
+Lemma ler_nimull x y : y ≤ 0 → x ≤ 1 → y ≤ x × y.
+ +
+Lemma ler_pimulr x y : 0 ≤ y → x ≤ 1 → y × x ≤ y.
+ +
+Lemma ler_nimulr 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).
+ +
+
+
++Lemma ler_pemull x y : 0 ≤ y → 1 ≤ x → y ≤ x × y.
+ +
+Lemma ler_nemull x y : y ≤ 0 → 1 ≤ x → x × y ≤ y.
+ +
+Lemma ler_pemulr x y : 0 ≤ y → 1 ≤ x → y ≤ y × x.
+ +
+Lemma ler_nemulr x y : y ≤ 0 → 1 ≤ x → y × x ≤ y.
+ +
+Lemma ler_pimull x y : 0 ≤ y → x ≤ 1 → x × y ≤ y.
+ +
+Lemma ler_nimull x y : y ≤ 0 → x ≤ 1 → y ≤ x × y.
+ +
+Lemma ler_pimulr x y : 0 ≤ y → x ≤ 1 → y × x ≤ y.
+ +
+Lemma ler_nimulr 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}.
+ +
+
+
++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%N).
+ +
+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%N).
+ +
+Definition exprn_egte1 := (exprn_ege1, exprn_egt1).
+Definition exprn_cp1 := (exprn_ilte1, exprn_egte1).
+ +
+Lemma ler_iexpr x n : (0 < n)%N → 0 ≤ x → x ≤ 1 → x ^+ n ≤ x.
+ +
+Lemma ltr_iexpr x n : 0 < x → x < 1 → (x ^+ n < x) = (1 < n)%N.
+ +
+Definition lter_iexpr := (ler_iexpr, ltr_iexpr).
+ +
+Lemma ler_eexpr x n : (0 < n)%N → 1 ≤ x → x ≤ x ^+ n.
+ +
+Lemma ltr_eexpr x n : 1 < x → (x < x ^+ n) = (1 < n)%N.
+ +
+Definition lter_eexpr := (ler_eexpr, ltr_eexpr).
+Definition lter_expr := (lter_iexpr, lter_eexpr).
+ +
+Lemma ler_wiexpn2l x :
+ 0 ≤ x → x ≤ 1 → {homo (GRing.exp x) : m n / (n ≤ m)%N >-> m ≤ n}.
+ +
+Lemma ler_weexpn2l 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%N) || (x == 1)).
+ +
+Lemma ieexprIn x : 0 < x → x != 1 → injective (GRing.exp x).
+ +
+Lemma ler_iexpn2l x :
+ 0 < x → x < 1 → {mono (GRing.exp x) : m n / (n ≤ m)%N >-> m ≤ n}.
+ +
+Lemma ltr_iexpn2l x :
+ 0 < x → x < 1 → {mono (GRing.exp x) : m n / (n < m)%N >-> m < n}.
+ +
+Definition lter_iexpn2l := (ler_iexpn2l, ltr_iexpn2l).
+ +
+Lemma ler_eexpn2l x :
+ 1 < x → {mono (GRing.exp x) : m n / (m ≤ n)%N >-> m ≤ n}.
+ +
+Lemma ltr_eexpn2l x :
+ 1 < x → {mono (GRing.exp x) : m n / (m < n)%N >-> m < n}.
+ +
+Definition lter_eexpn2l := (ler_eexpn2l, ltr_eexpn2l).
+ +
+Lemma ltr_expn2r n x y : 0 ≤ x → x < y → x ^+ n < y ^+ n = (n != 0%N).
+ +
+Lemma ler_expn2r n : {in nneg & , {homo ((@GRing.exp R)^~ n) : x y / x ≤ y}}.
+ +
+Definition lter_expn2r := (ler_expn2r, ltr_expn2r).
+ +
+Lemma ltr_wpexpn2r n :
+ (0 < n)%N → {in nneg & , {homo ((@GRing.exp R)^~ n) : x y / x < y}}.
+ +
+Lemma ler_pexpn2r n :
+ (0 < n)%N → {in nneg & , {mono ((@GRing.exp R)^~ n) : x y / x ≤ y}}.
+ +
+Lemma ltr_pexpn2r n :
+ (0 < n)%N → {in nneg & , {mono ((@GRing.exp R)^~ n) : x y / x < y}}.
+ +
+Definition lter_pexpn2r := (ler_pexpn2r, ltr_pexpn2r).
+ +
+Lemma pexpIrn n : (0 < n)%N → {in nneg &, injective ((@GRing.exp R)^~ 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%N).
+ +
+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%N).
+ +
+Definition exprn_egte1 := (exprn_ege1, exprn_egt1).
+Definition exprn_cp1 := (exprn_ilte1, exprn_egte1).
+ +
+Lemma ler_iexpr x n : (0 < n)%N → 0 ≤ x → x ≤ 1 → x ^+ n ≤ x.
+ +
+Lemma ltr_iexpr x n : 0 < x → x < 1 → (x ^+ n < x) = (1 < n)%N.
+ +
+Definition lter_iexpr := (ler_iexpr, ltr_iexpr).
+ +
+Lemma ler_eexpr x n : (0 < n)%N → 1 ≤ x → x ≤ x ^+ n.
+ +
+Lemma ltr_eexpr x n : 1 < x → (x < x ^+ n) = (1 < n)%N.
+ +
+Definition lter_eexpr := (ler_eexpr, ltr_eexpr).
+Definition lter_expr := (lter_iexpr, lter_eexpr).
+ +
+Lemma ler_wiexpn2l x :
+ 0 ≤ x → x ≤ 1 → {homo (GRing.exp x) : m n / (n ≤ m)%N >-> m ≤ n}.
+ +
+Lemma ler_weexpn2l 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%N) || (x == 1)).
+ +
+Lemma ieexprIn x : 0 < x → x != 1 → injective (GRing.exp x).
+ +
+Lemma ler_iexpn2l x :
+ 0 < x → x < 1 → {mono (GRing.exp x) : m n / (n ≤ m)%N >-> m ≤ n}.
+ +
+Lemma ltr_iexpn2l x :
+ 0 < x → x < 1 → {mono (GRing.exp x) : m n / (n < m)%N >-> m < n}.
+ +
+Definition lter_iexpn2l := (ler_iexpn2l, ltr_iexpn2l).
+ +
+Lemma ler_eexpn2l x :
+ 1 < x → {mono (GRing.exp x) : m n / (m ≤ n)%N >-> m ≤ n}.
+ +
+Lemma ltr_eexpn2l x :
+ 1 < x → {mono (GRing.exp x) : m n / (m < n)%N >-> m < n}.
+ +
+Definition lter_eexpn2l := (ler_eexpn2l, ltr_eexpn2l).
+ +
+Lemma ltr_expn2r n x y : 0 ≤ x → x < y → x ^+ n < y ^+ n = (n != 0%N).
+ +
+Lemma ler_expn2r n : {in nneg & , {homo ((@GRing.exp R)^~ n) : x y / x ≤ y}}.
+ +
+Definition lter_expn2r := (ler_expn2r, ltr_expn2r).
+ +
+Lemma ltr_wpexpn2r n :
+ (0 < n)%N → {in nneg & , {homo ((@GRing.exp R)^~ n) : x y / x < y}}.
+ +
+Lemma ler_pexpn2r n :
+ (0 < n)%N → {in nneg & , {mono ((@GRing.exp R)^~ n) : x y / x ≤ y}}.
+ +
+Lemma ltr_pexpn2r n :
+ (0 < n)%N → {in nneg & , {mono ((@GRing.exp R)^~ n) : x y / x < y}}.
+ +
+Definition lter_pexpn2r := (ler_pexpn2r, ltr_pexpn2r).
+ +
+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%N) || (x == 1).
+ +
+Lemma eqr_expn2 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_pinv :
+ {in [pred x in GRing.unit | 0 < x] &, {mono (@GRing.inv R) : x y /~ x ≤ y}}.
+ +
+Lemma ler_ninv :
+ {in [pred x in GRing.unit | x < 0] &, {mono (@GRing.inv R) : x y /~ x ≤ y}}.
+ +
+Lemma ltr_pinv :
+ {in [pred x in GRing.unit | 0 < x] &, {mono (@GRing.inv R) : x y /~ x < y}}.
+ +
+Lemma ltr_ninv :
+ {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).
+ +
+
+
++ +
+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%N) || (x == 1).
+ +
+Lemma eqr_expn2 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_pinv :
+ {in [pred x in GRing.unit | 0 < x] &, {mono (@GRing.inv R) : x y /~ x ≤ y}}.
+ +
+Lemma ler_ninv :
+ {in [pred x in GRing.unit | x < 0] &, {mono (@GRing.inv R) : x y /~ x ≤ y}}.
+ +
+Lemma ltr_pinv :
+ {in [pred x in GRing.unit | 0 < x] &, {mono (@GRing.inv R) : x y /~ x < y}}.
+ +
+Lemma ltr_ninv :
+ {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).
+ +
+
+ norm
+
+
+
+
+ norm + add
+
+
+
+
+Lemma normr_real x : `|x| \is real.
+Hint Resolve normr_real.
+ +
+Lemma ler_norm_sum I r (G : I → R) (P : pred I):
+ `|\sum_(i <- r | P i) G i| ≤ \sum_(i <- r | P i) `|G i|.
+ +
+Lemma ler_norm_sub x y : `|x - y| ≤ `|x| + `|y|.
+ +
+Lemma ler_dist_add z x y : `|x - y| ≤ `|x - z| + `|z - y|.
+ +
+Lemma ler_sub_norm_add x y : `|x| - `|y| ≤ `|x + y|.
+ +
+Lemma ler_sub_dist x y : `|x| - `|y| ≤ `|x - y|.
+ +
+Lemma ler_dist_dist x y : `|`|x| - `|y| | ≤ `|x - y|.
+ +
+Lemma ler_dist_norm_add x y : `| `|x| - `|y| | ≤ `| x + y |.
+ +
+Lemma real_ler_norml x y : x \is real → (`|x| ≤ y) = (- y ≤ x ≤ y).
+ +
+Lemma real_ler_normlP x y :
+ x \is real → reflect ((-x ≤ y) × (x ≤ y)) (`|x| ≤ y).
+ +
+Lemma real_eqr_norml x y :
+ x \is real → (`|x| == y) = ((x == y) || (x == -y)) && (0 ≤ y).
+ +
+Lemma real_eqr_norm2 x y :
+ x \is real → y \is real → (`|x| == `|y|) = (x == y) || (x == -y).
+ +
+Lemma real_ltr_norml x y : x \is real → (`|x| < y) = (- y < x < y).
+ +
+Definition real_lter_norml := (real_ler_norml, real_ltr_norml).
+ +
+Lemma real_ltr_normlP x y :
+ x \is real → reflect ((-x < y) × (x < y)) (`|x| < y).
+ +
+Lemma real_ler_normr x y : y \is real → (x ≤ `|y|) = (x ≤ y) || (x ≤ - y).
+ +
+Lemma real_ltr_normr x y : y \is real → (x < `|y|) = (x < y) || (x < - y).
+ +
+Definition real_lter_normr := (real_ler_normr, real_ltr_normr).
+ +
+Lemma ler_nnorml x y : y < 0 → `|x| ≤ y = false.
+ +
+Lemma ltr_nnorml x y : y ≤ 0 → `|x| < y = false.
+ +
+Definition lter_nnormr := (ler_nnorml, ltr_nnorml).
+ +
+Lemma real_ler_distl x y e :
+ x - y \is real → (`|x - y| ≤ e) = (y - e ≤ x ≤ y + e).
+ +
+Lemma real_ltr_distl x y e :
+ x - y \is real → (`|x - y| < e) = (y - e < x < y + e).
+ +
+Definition real_lter_distl := (real_ler_distl, real_ltr_distl).
+ +
+(* GG: pointless duplication }-( *)
+Lemma eqr_norm_id x : (`|x| == x) = (0 ≤ x).
+Lemma eqr_normN x : (`|x| == - x) = (x ≤ 0).
+Definition eqr_norm_idVN := =^~ (ger0_def, ler0_def).
+ +
+Lemma real_exprn_even_ge0 n x : x \is real → ~~ odd n → 0 ≤ x ^+ n.
+ +
+Lemma real_exprn_even_gt0 n x :
+ x \is real → ~~ odd n → (0 < x ^+ n) = (n == 0)%N || (x != 0).
+ +
+Lemma real_exprn_even_le0 n x :
+ x \is real → ~~ odd n → (x ^+ n ≤ 0) = (n != 0%N) && (x == 0).
+ +
+Lemma real_exprn_even_lt0 n x :
+ x \is real → ~~ odd n → (x ^+ n < 0) = false.
+ +
+Lemma real_exprn_odd_ge0 n x :
+ x \is real → odd n → (0 ≤ x ^+ n) = (0 ≤ x).
+ +
+Lemma real_exprn_odd_gt0 n x : x \is real → odd n → (0 < x ^+ n) = (0 < x).
+ +
+Lemma real_exprn_odd_le0 n x : x \is real → odd n → (x ^+ n ≤ 0) = (x ≤ 0).
+ +
+Lemma real_exprn_odd_lt0 n x : x \is real → odd n → (x ^+ n < 0) = (x < 0).
+ +
+
+
++Lemma normr_real x : `|x| \is real.
+Hint Resolve normr_real.
+ +
+Lemma ler_norm_sum I r (G : I → R) (P : pred I):
+ `|\sum_(i <- r | P i) G i| ≤ \sum_(i <- r | P i) `|G i|.
+ +
+Lemma ler_norm_sub x y : `|x - y| ≤ `|x| + `|y|.
+ +
+Lemma ler_dist_add z x y : `|x - y| ≤ `|x - z| + `|z - y|.
+ +
+Lemma ler_sub_norm_add x y : `|x| - `|y| ≤ `|x + y|.
+ +
+Lemma ler_sub_dist x y : `|x| - `|y| ≤ `|x - y|.
+ +
+Lemma ler_dist_dist x y : `|`|x| - `|y| | ≤ `|x - y|.
+ +
+Lemma ler_dist_norm_add x y : `| `|x| - `|y| | ≤ `| x + y |.
+ +
+Lemma real_ler_norml x y : x \is real → (`|x| ≤ y) = (- y ≤ x ≤ y).
+ +
+Lemma real_ler_normlP x y :
+ x \is real → reflect ((-x ≤ y) × (x ≤ y)) (`|x| ≤ y).
+ +
+Lemma real_eqr_norml x y :
+ x \is real → (`|x| == y) = ((x == y) || (x == -y)) && (0 ≤ y).
+ +
+Lemma real_eqr_norm2 x y :
+ x \is real → y \is real → (`|x| == `|y|) = (x == y) || (x == -y).
+ +
+Lemma real_ltr_norml x y : x \is real → (`|x| < y) = (- y < x < y).
+ +
+Definition real_lter_norml := (real_ler_norml, real_ltr_norml).
+ +
+Lemma real_ltr_normlP x y :
+ x \is real → reflect ((-x < y) × (x < y)) (`|x| < y).
+ +
+Lemma real_ler_normr x y : y \is real → (x ≤ `|y|) = (x ≤ y) || (x ≤ - y).
+ +
+Lemma real_ltr_normr x y : y \is real → (x < `|y|) = (x < y) || (x < - y).
+ +
+Definition real_lter_normr := (real_ler_normr, real_ltr_normr).
+ +
+Lemma ler_nnorml x y : y < 0 → `|x| ≤ y = false.
+ +
+Lemma ltr_nnorml x y : y ≤ 0 → `|x| < y = false.
+ +
+Definition lter_nnormr := (ler_nnorml, ltr_nnorml).
+ +
+Lemma real_ler_distl x y e :
+ x - y \is real → (`|x - y| ≤ e) = (y - e ≤ x ≤ y + e).
+ +
+Lemma real_ltr_distl x y e :
+ x - y \is real → (`|x - y| < e) = (y - e < x < y + e).
+ +
+Definition real_lter_distl := (real_ler_distl, real_ltr_distl).
+ +
+(* GG: pointless duplication }-( *)
+Lemma eqr_norm_id x : (`|x| == x) = (0 ≤ x).
+Lemma eqr_normN x : (`|x| == - x) = (x ≤ 0).
+Definition eqr_norm_idVN := =^~ (ger0_def, ler0_def).
+ +
+Lemma real_exprn_even_ge0 n x : x \is real → ~~ odd n → 0 ≤ x ^+ n.
+ +
+Lemma real_exprn_even_gt0 n x :
+ x \is real → ~~ odd n → (0 < x ^+ n) = (n == 0)%N || (x != 0).
+ +
+Lemma real_exprn_even_le0 n x :
+ x \is real → ~~ odd n → (x ^+ n ≤ 0) = (n != 0%N) && (x == 0).
+ +
+Lemma real_exprn_even_lt0 n x :
+ x \is real → ~~ odd n → (x ^+ n < 0) = false.
+ +
+Lemma real_exprn_odd_ge0 n x :
+ x \is real → odd n → (0 ≤ x ^+ n) = (0 ≤ x).
+ +
+Lemma real_exprn_odd_gt0 n x : x \is real → odd n → (0 < x ^+ n) = (0 < x).
+ +
+Lemma real_exprn_odd_le0 n x : x \is real → odd n → (x ^+ n ≤ 0) = (x ≤ 0).
+ +
+Lemma real_exprn_odd_lt0 n x : x \is real → odd n → (x ^+ n < 0) = (x < 0).
+ +
+
+ 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.
+ +
+
+
++ +
+Lemma real_normK x : x \is real → `|x| ^+ 2 = x ^+ 2.
+ +
+
+ Binary sign ((-1) ^+ s).
+
+
+
+
+Lemma normr_sign s : `|(-1) ^+ s| = 1 :> R.
+ +
+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.
+ +
+
+
++Lemma normr_sign s : `|(-1) ^+ s| = 1 :> R.
+ +
+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.
+
+
+
+
+ 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%N)%:R × sg x.
+ +
+Lemma sgr_nat n : sg n%:R = (n != 0%N)%: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).
+ +
+
+
++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%N)%:R × sg x.
+ +
+Lemma sgr_nat n : sg n%:R = (n != 0%N)%: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.
+ +
+
+
++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.
+ +
+
+ GG: pointless duplication...
+
+
+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.
+ +
+
+
++ +
+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.
+ +
+
+ lerif
+
+
+
+
+Lemma lerif_refl x C : reflect (x ≤ x ?= iff C) C.
+ +
+Lemma lerif_trans x1 x2 x3 C12 C23 :
+ x1 ≤ x2 ?= iff C12 → x2 ≤ x3 ?= iff C23 → x1 ≤ x3 ?= iff C12 && C23.
+ +
+Lemma lerif_le x y : x ≤ y → x ≤ y ?= iff (x ≥ y).
+ +
+Lemma lerif_eq x y : x ≤ y → x ≤ y ?= iff (x == y).
+ +
+Lemma ger_lerif x y C : x ≤ y ?= iff C → (y ≤ x) = C.
+ +
+Lemma ltr_lerif x y C : x ≤ y ?= iff C → (x < y) = ~~ C.
+ +
+Lemma lerif_nat m n C : (m%:R ≤ n%:R ?= iff C :> R) = (m ≤ n ?= iff C)%N.
+ +
+Lemma mono_in_lerif (A : pred R) (f : R → R) C :
+ {in A &, {mono f : x y / x ≤ y}} →
+ {in A &, ∀ x y, (f x ≤ f y ?= iff C) = (x ≤ y ?= iff C)}.
+ +
+Lemma mono_lerif (f : R → R) C :
+ {mono f : x y / x ≤ y} →
+ ∀ x y, (f x ≤ f y ?= iff C) = (x ≤ y ?= iff C).
+ +
+Lemma nmono_in_lerif (A : pred R) (f : R → R) C :
+ {in A &, {mono f : x y /~ x ≤ y}} →
+ {in A &, ∀ x y, (f x ≤ f y ?= iff C) = (y ≤ x ?= iff C)}.
+ +
+Lemma nmono_lerif (f : R → R) C :
+ {mono f : x y /~ x ≤ y} →
+ ∀ x y, (f x ≤ f y ?= iff C) = (y ≤ x ?= iff C).
+ +
+Lemma lerif_subLR x y z C : (x - y ≤ z ?= iff C) = (x ≤ z + y ?= iff C).
+ +
+Lemma lerif_subRL x y z C : (x ≤ y - z ?= iff C) = (x + z ≤ y ?= iff C).
+ +
+Lemma lerif_add x1 y1 C1 x2 y2 C2 :
+ x1 ≤ y1 ?= iff C1 → x2 ≤ y2 ?= iff C2 →
+ x1 + x2 ≤ y1 + y2 ?= iff C1 && C2.
+ +
+Lemma lerif_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 lerif_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_lerif_norm x : x \is real → x ≤ `|x| ?= iff (0 ≤ x).
+ +
+Lemma lerif_pmul 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 lerif_nmul 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 lerif_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].
+ +
+
+
++Lemma lerif_refl x C : reflect (x ≤ x ?= iff C) C.
+ +
+Lemma lerif_trans x1 x2 x3 C12 C23 :
+ x1 ≤ x2 ?= iff C12 → x2 ≤ x3 ?= iff C23 → x1 ≤ x3 ?= iff C12 && C23.
+ +
+Lemma lerif_le x y : x ≤ y → x ≤ y ?= iff (x ≥ y).
+ +
+Lemma lerif_eq x y : x ≤ y → x ≤ y ?= iff (x == y).
+ +
+Lemma ger_lerif x y C : x ≤ y ?= iff C → (y ≤ x) = C.
+ +
+Lemma ltr_lerif x y C : x ≤ y ?= iff C → (x < y) = ~~ C.
+ +
+Lemma lerif_nat m n C : (m%:R ≤ n%:R ?= iff C :> R) = (m ≤ n ?= iff C)%N.
+ +
+Lemma mono_in_lerif (A : pred R) (f : R → R) C :
+ {in A &, {mono f : x y / x ≤ y}} →
+ {in A &, ∀ x y, (f x ≤ f y ?= iff C) = (x ≤ y ?= iff C)}.
+ +
+Lemma mono_lerif (f : R → R) C :
+ {mono f : x y / x ≤ y} →
+ ∀ x y, (f x ≤ f y ?= iff C) = (x ≤ y ?= iff C).
+ +
+Lemma nmono_in_lerif (A : pred R) (f : R → R) C :
+ {in A &, {mono f : x y /~ x ≤ y}} →
+ {in A &, ∀ x y, (f x ≤ f y ?= iff C) = (y ≤ x ?= iff C)}.
+ +
+Lemma nmono_lerif (f : R → R) C :
+ {mono f : x y /~ x ≤ y} →
+ ∀ x y, (f x ≤ f y ?= iff C) = (y ≤ x ?= iff C).
+ +
+Lemma lerif_subLR x y z C : (x - y ≤ z ?= iff C) = (x ≤ z + y ?= iff C).
+ +
+Lemma lerif_subRL x y z C : (x ≤ y - z ?= iff C) = (x + z ≤ y ?= iff C).
+ +
+Lemma lerif_add x1 y1 C1 x2 y2 C2 :
+ x1 ≤ y1 ?= iff C1 → x2 ≤ y2 ?= iff C2 →
+ x1 + x2 ≤ y1 + y2 ?= iff C1 && C2.
+ +
+Lemma lerif_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 lerif_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_lerif_norm x : x \is real → x ≤ `|x| ?= iff (0 ≤ x).
+ +
+Lemma lerif_pmul 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 lerif_nmul 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 lerif_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].
+ +
+
+ Mean inequalities.
+
+
+
+
+Lemma real_lerif_mean_square_scaled x y :
+ x \is real → y \is real → x × y *+ 2 ≤ x ^+ 2 + y ^+ 2 ?= iff (x == y).
+ +
+Lemma real_lerif_AGM2_scaled x y :
+ x \is real → y \is real → x × y *+ 4 ≤ (x + y) ^+ 2 ?= iff (x == y).
+ +
+Lemma lerif_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].
+ +
+
+
++Lemma real_lerif_mean_square_scaled x y :
+ x \is real → y \is real → x × y *+ 2 ≤ x ^+ 2 + y ^+ 2 ?= iff (x == y).
+ +
+Lemma real_lerif_AGM2_scaled x y :
+ x \is real → y \is real → x × y *+ 4 ≤ (x + y) ^+ 2 ?= iff (x == y).
+ +
+Lemma lerif_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.
+ +
+Hint Resolve ler_opp2 ltr_opp2 real0 real1 normr_real.
+ +
+Section NumDomainMonotonyTheoryForReals.
+ +
+Variables (R R' : numDomainType) (D : pred R) (f : R → R').
+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}}.
+ +
+
+
++Implicit Type p : {poly R}.
+ +
+Lemma poly_disk_bound p b : {ub | ∀ x, `|x| ≤ b → `|p.[x]| ≤ ub}.
+ +
+End NumDomainOperationTheory.
+ +
+Hint Resolve ler_opp2 ltr_opp2 real0 real1 normr_real.
+ +
+Section NumDomainMonotonyTheoryForReals.
+ +
+Variables (R R' : numDomainType) (D : pred R) (f : R → R').
+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}}.
+ +
+
+ GG: Domain should precede condition.
+
+
+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}}.
+ +
+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_pinv : {in pos &, {mono (@GRing.inv F) : x y /~ x ≤ y}}.
+ +
+Lemma lef_ninv : {in neg &, {mono (@GRing.inv F) : x y /~ x ≤ y}}.
+ +
+Lemma ltf_pinv : {in pos &, {mono (@GRing.inv F) : x y /~ x < y}}.
+ +
+Lemma ltf_ninv: {in neg &, {mono (@GRing.inv F) : x y /~ x < y}}.
+ +
+Definition ltef_pinv := (lef_pinv, ltf_pinv).
+Definition ltef_ninv := (lef_ninv, ltf_ninv).
+ +
+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_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).
+ +
+
+
++ {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}}.
+ +
+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_pinv : {in pos &, {mono (@GRing.inv F) : x y /~ x ≤ y}}.
+ +
+Lemma lef_ninv : {in neg &, {mono (@GRing.inv F) : x y /~ x ≤ y}}.
+ +
+Lemma ltf_pinv : {in pos &, {mono (@GRing.inv F) : x y /~ x < y}}.
+ +
+Lemma ltf_ninv: {in neg &, {mono (@GRing.inv F) : x y /~ x < y}}.
+ +
+Definition ltef_pinv := (lef_pinv, ltf_pinv).
+Definition ltef_ninv := (lef_ninv, ltf_ninv).
+ +
+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_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_pdivl_mulr z x y : 0 < z → (x ≤ y / z) = (x × z ≤ y).
+ +
+Lemma ltr_pdivl_mulr z x y : 0 < z → (x < y / z) = (x × z < y).
+ +
+Definition lter_pdivl_mulr := (ler_pdivl_mulr, ltr_pdivl_mulr).
+ +
+Lemma ler_pdivr_mulr z x y : 0 < z → (y / z ≤ x) = (y ≤ x × z).
+ +
+Lemma ltr_pdivr_mulr z x y : 0 < z → (y / z < x) = (y < x × z).
+ +
+Definition lter_pdivr_mulr := (ler_pdivr_mulr, ltr_pdivr_mulr).
+ +
+Lemma ler_pdivl_mull z x y : 0 < z → (x ≤ z^-1 × y) = (z × x ≤ y).
+ +
+Lemma ltr_pdivl_mull z x y : 0 < z → (x < z^-1 × y) = (z × x < y).
+ +
+Definition lter_pdivl_mull := (ler_pdivl_mull, ltr_pdivl_mull).
+ +
+Lemma ler_pdivr_mull z x y : 0 < z → (z^-1 × y ≤ x) = (y ≤ z × x).
+ +
+Lemma ltr_pdivr_mull z x y : 0 < z → (z^-1 × y < x) = (y < z × x).
+ +
+Definition lter_pdivr_mull := (ler_pdivr_mull, ltr_pdivr_mull).
+ +
+Lemma ler_ndivl_mulr z x y : z < 0 → (x ≤ y / z) = (y ≤ x × z).
+ +
+Lemma ltr_ndivl_mulr z x y : z < 0 → (x < y / z) = (y < x × z).
+ +
+Definition lter_ndivl_mulr := (ler_ndivl_mulr, ltr_ndivl_mulr).
+ +
+Lemma ler_ndivr_mulr z x y : z < 0 → (y / z ≤ x) = (x × z ≤ y).
+ +
+Lemma ltr_ndivr_mulr z x y : z < 0 → (y / z < x) = (x × z < y).
+ +
+Definition lter_ndivr_mulr := (ler_ndivr_mulr, ltr_ndivr_mulr).
+ +
+Lemma ler_ndivl_mull z x y : z < 0 → (x ≤ z^-1 × y) = (y ≤ z × x).
+ +
+Lemma ltr_ndivl_mull z x y : z < 0 → (x < z^-1 × y) = (y < z × x).
+ +
+Definition lter_ndivl_mull := (ler_ndivl_mull, ltr_ndivl_mull).
+ +
+Lemma ler_ndivr_mull z x y : z < 0 → (z^-1 × y ≤ x) = (z × x ≤ y).
+ +
+Lemma ltr_ndivr_mull z x y : z < 0 → (z^-1 × y < x) = (z × x < y).
+ +
+Definition lter_ndivr_mull := (ler_ndivr_mull, ltr_ndivr_mull).
+ +
+Lemma natf_div m d : (d %| m)%N → (m %/ d)%:R = m%:R / d%:R :> F.
+ +
+Lemma normfV : {morph (@norm F) : x / x ^-1}.
+ +
+Lemma normf_div : {morph (@norm F) : x y / x / y}.
+ +
+Lemma invr_sg x : (sg x)^-1 = sgr x.
+ +
+Lemma sgrV x : sgr x^-1 = sgr x.
+ +
+
+
++ +
+Lemma ltr_pdivl_mulr z x y : 0 < z → (x < y / z) = (x × z < y).
+ +
+Definition lter_pdivl_mulr := (ler_pdivl_mulr, ltr_pdivl_mulr).
+ +
+Lemma ler_pdivr_mulr z x y : 0 < z → (y / z ≤ x) = (y ≤ x × z).
+ +
+Lemma ltr_pdivr_mulr z x y : 0 < z → (y / z < x) = (y < x × z).
+ +
+Definition lter_pdivr_mulr := (ler_pdivr_mulr, ltr_pdivr_mulr).
+ +
+Lemma ler_pdivl_mull z x y : 0 < z → (x ≤ z^-1 × y) = (z × x ≤ y).
+ +
+Lemma ltr_pdivl_mull z x y : 0 < z → (x < z^-1 × y) = (z × x < y).
+ +
+Definition lter_pdivl_mull := (ler_pdivl_mull, ltr_pdivl_mull).
+ +
+Lemma ler_pdivr_mull z x y : 0 < z → (z^-1 × y ≤ x) = (y ≤ z × x).
+ +
+Lemma ltr_pdivr_mull z x y : 0 < z → (z^-1 × y < x) = (y < z × x).
+ +
+Definition lter_pdivr_mull := (ler_pdivr_mull, ltr_pdivr_mull).
+ +
+Lemma ler_ndivl_mulr z x y : z < 0 → (x ≤ y / z) = (y ≤ x × z).
+ +
+Lemma ltr_ndivl_mulr z x y : z < 0 → (x < y / z) = (y < x × z).
+ +
+Definition lter_ndivl_mulr := (ler_ndivl_mulr, ltr_ndivl_mulr).
+ +
+Lemma ler_ndivr_mulr z x y : z < 0 → (y / z ≤ x) = (x × z ≤ y).
+ +
+Lemma ltr_ndivr_mulr z x y : z < 0 → (y / z < x) = (x × z < y).
+ +
+Definition lter_ndivr_mulr := (ler_ndivr_mulr, ltr_ndivr_mulr).
+ +
+Lemma ler_ndivl_mull z x y : z < 0 → (x ≤ z^-1 × y) = (y ≤ z × x).
+ +
+Lemma ltr_ndivl_mull z x y : z < 0 → (x < z^-1 × y) = (y < z × x).
+ +
+Definition lter_ndivl_mull := (ler_ndivl_mull, ltr_ndivl_mull).
+ +
+Lemma ler_ndivr_mull z x y : z < 0 → (z^-1 × y ≤ x) = (z × x ≤ y).
+ +
+Lemma ltr_ndivr_mull z x y : z < 0 → (z^-1 × y < x) = (z × x < y).
+ +
+Definition lter_ndivr_mull := (ler_ndivr_mull, ltr_ndivr_mull).
+ +
+Lemma natf_div m d : (d %| m)%N → (m %/ d)%:R = m%:R / d%:R :> F.
+ +
+Lemma normfV : {morph (@norm F) : x / x ^-1}.
+ +
+Lemma normf_div : {morph (@norm F) : x y / x / y}.
+ +
+Lemma invr_sg x : (sg x)^-1 = sgr x.
+ +
+Lemma sgrV x : sgr x^-1 = sgr x.
+ +
+
+ Interval midpoint.
+
+
+
+
+ +
+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 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).
+ +
+
+ The AGM, unscaled but without the nth root.
+
+
+
+
+Lemma real_lerif_mean_square x y :
+ x \is real → y \is real → x × y ≤ mid (x ^+ 2) (y ^+ 2) ?= iff (x == y).
+ +
+Lemma real_lerif_AGM2 x y :
+ x \is real → y \is real → x × y ≤ mid x y ^+ 2 ?= iff (x == y).
+ +
+Lemma lerif_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.
+ +
+Hint Resolve lerr.
+ +
+Variable R : realDomainType.
+Implicit Types x y z t : R.
+ +
+Lemma num_real x : x \is real.
+Hint Resolve num_real.
+ +
+Lemma ler_total : total (@le R).
+ +
+Lemma ltr_total x y : x != y → (x < y) || (y < x).
+ +
+Lemma wlog_ler P :
+ (∀ a b, P b a → P a b) → (∀ a b, a ≤ b → P a b) →
+ ∀ a b : R, P a b.
+ +
+Lemma 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, P a b.
+ +
+Lemma ltrNge x y : (x < y) = ~~ (y ≤ x).
+ +
+Lemma lerNgt x y : (x ≤ y) = ~~ (y < x).
+ +
+Lemma lerP x y : ler_xor_gt x y `|x - y| `|y - x| (x ≤ y) (y < x).
+ +
+Lemma ltrP x y : ltr_xor_ge x y `|x - y| `|y - x| (y ≤ x) (x < y).
+ +
+Lemma ltrgtP x y :
+ comparer x y `|x - y| `|y - x| (y == x) (x == y)
+ (x ≤ y) (y ≤ x) (x < y) (x > y) .
+ +
+Lemma ger0P x : ger0_xor_lt0 x `|x| (x < 0) (0 ≤ x).
+ +
+Lemma ler0P x : ler0_xor_gt0 x `|x| (0 < x) (x ≤ 0).
+ +
+Lemma ltrgt0P x :
+ comparer0 x `|x| (0 == x) (x == 0) (x ≤ 0) (0 ≤ x) (x < 0) (x > 0).
+ +
+Lemma neqr_lt x y : (x != y) = (x < y) || (y < x).
+ +
+Lemma eqr_leLR x y z t :
+ (x ≤ y → z ≤ t) → (y < x → t < z) → (x ≤ y) = (z ≤ t).
+ +
+Lemma eqr_leRL x y z t :
+ (x ≤ y → z ≤ t) → (y < x → t < z) → (z ≤ t) = (x ≤ y).
+ +
+Lemma eqr_ltLR x y z t :
+ (x < y → z < t) → (y ≤ x → t ≤ z) → (x < y) = (z < t).
+ +
+Lemma eqr_ltRL x y z t :
+ (x < y → z < t) → (y ≤ x → t ≤ z) → (z < t) = (x < y).
+ +
+
+
++Lemma real_lerif_mean_square x y :
+ x \is real → y \is real → x × y ≤ mid (x ^+ 2) (y ^+ 2) ?= iff (x == y).
+ +
+Lemma real_lerif_AGM2 x y :
+ x \is real → y \is real → x × y ≤ mid x y ^+ 2 ?= iff (x == y).
+ +
+Lemma lerif_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.
+ +
+Hint Resolve lerr.
+ +
+Variable R : realDomainType.
+Implicit Types x y z t : R.
+ +
+Lemma num_real x : x \is real.
+Hint Resolve num_real.
+ +
+Lemma ler_total : total (@le R).
+ +
+Lemma ltr_total x y : x != y → (x < y) || (y < x).
+ +
+Lemma wlog_ler P :
+ (∀ a b, P b a → P a b) → (∀ a b, a ≤ b → P a b) →
+ ∀ a b : R, P a b.
+ +
+Lemma 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, P a b.
+ +
+Lemma ltrNge x y : (x < y) = ~~ (y ≤ x).
+ +
+Lemma lerNgt x y : (x ≤ y) = ~~ (y < x).
+ +
+Lemma lerP x y : ler_xor_gt x y `|x - y| `|y - x| (x ≤ y) (y < x).
+ +
+Lemma ltrP x y : ltr_xor_ge x y `|x - y| `|y - x| (y ≤ x) (x < y).
+ +
+Lemma ltrgtP x y :
+ comparer x y `|x - y| `|y - x| (y == x) (x == y)
+ (x ≤ y) (y ≤ x) (x < y) (x > y) .
+ +
+Lemma ger0P x : ger0_xor_lt0 x `|x| (x < 0) (0 ≤ x).
+ +
+Lemma ler0P x : ler0_xor_gt0 x `|x| (0 < x) (x ≤ 0).
+ +
+Lemma ltrgt0P x :
+ comparer0 x `|x| (0 == x) (x == 0) (x ≤ 0) (0 ≤ x) (x < 0) (x > 0).
+ +
+Lemma neqr_lt x y : (x != y) = (x < y) || (y < x).
+ +
+Lemma eqr_leLR x y z t :
+ (x ≤ y → z ≤ t) → (y < x → t < z) → (x ≤ y) = (z ≤ t).
+ +
+Lemma eqr_leRL x y z t :
+ (x ≤ y → z ≤ t) → (y < x → t < z) → (z ≤ t) = (x ≤ y).
+ +
+Lemma eqr_ltLR x y z t :
+ (x < y → z < t) → (y ≤ x → t ≤ z) → (x < y) = (z < t).
+ +
+Lemma eqr_ltRL x y z t :
+ (x < y → z < t) → (y ≤ x → t ≤ z) → (z < t) = (x < y).
+ +
+
+ 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).
+ +
+
+
++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.
+ +
+Hint Resolve num_real.
+ +
+Section RealDomainMonotony.
+ +
+Variables (R : realDomainType) (R' : numDomainType) (D : pred R) (f : R → R').
+Implicit Types (m n p : nat) (x y z : R) (u v w : R').
+ +
+Hint Resolve (@num_real R).
+ +
+Lemma homo_mono : {homo f : x y / x < y} → {mono f : x y / x ≤ y}.
+ +
+Lemma nhomo_mono : {homo f : x y /~ x < y} → {mono f : x y /~ x ≤ y}.
+ +
+Lemma homo_mono_in :
+ {in D &, {homo f : x y / x < y}} → {in D &, {mono f : x y / x ≤ y}}.
+ +
+Lemma nhomo_mono_in :
+ {in D &, {homo f : x y /~ x < y}} → {in D &, {mono f : x y /~ x ≤ y}}.
+ +
+End RealDomainMonotony.
+ +
+Section RealDomainOperations.
+ +
+
+
++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.
+ +
+Hint Resolve num_real.
+ +
+Section RealDomainMonotony.
+ +
+Variables (R : realDomainType) (R' : numDomainType) (D : pred R) (f : R → R').
+Implicit Types (m n p : nat) (x y z : R) (u v w : R').
+ +
+Hint Resolve (@num_real R).
+ +
+Lemma homo_mono : {homo f : x y / x < y} → {mono f : x y / x ≤ y}.
+ +
+Lemma nhomo_mono : {homo f : x y /~ x < y} → {mono f : x y /~ x ≤ y}.
+ +
+Lemma homo_mono_in :
+ {in D &, {homo f : x y / x < y}} → {in D &, {mono f : x y / x ≤ y}}.
+ +
+Lemma nhomo_mono_in :
+ {in D &, {homo f : x y /~ x < y}} → {in D &, {mono f : x y /~ x ≤ y}}.
+ +
+End RealDomainMonotony.
+ +
+Section RealDomainOperations.
+ +
+
+ sgr section
+
+
+
+
+Variable R : realDomainType.
+Implicit Types x y z t : R.
+Hint Resolve (@num_real R).
+ +
+Lemma sgr_cp0 x :
+ ((sg x == 1) = (0 < x)) ×
+ ((sg x == -1) = (x < 0)) ×
+ ((sg x == 0) = (x == 0)).
+ +
+CoInductive 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|.
+ +
+
+
++Variable R : realDomainType.
+Implicit Types x y z t : R.
+Hint Resolve (@num_real R).
+ +
+Lemma sgr_cp0 x :
+ ((sg x == 1) = (0 < x)) ×
+ ((sg x == -1) = (x < 0)) ×
+ ((sg x == 0) = (x == 0)).
+ +
+CoInductive 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|.
+ +
+
+ GG: duplicate!
+
+
+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).
+ +
+
+
++ +
+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).
+ +
+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).
+ +
+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 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%N) && (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).
+ +
+
+
++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).
+ +
+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).
+ +
+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 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%N) && (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).
+ +
+
+ Special lemmas for squares.
+
+
+
+
+Lemma sqr_ge0 x : 0 ≤ x ^+ 2.
+ +
+Lemma sqr_norm_eq1 x : (x ^+ 2 == 1) = (`|x| == 1).
+ +
+Lemma lerif_mean_square_scaled x y :
+ x × y *+ 2 ≤ x ^+ 2 + y ^+ 2 ?= iff (x == y).
+ +
+Lemma lerif_AGM2_scaled x y : x × y *+ 4 ≤ (x + y) ^+ 2 ?= iff (x == y).
+ +
+Section MinMax.
+ +
+
+
++Lemma sqr_ge0 x : 0 ≤ x ^+ 2.
+ +
+Lemma sqr_norm_eq1 x : (x ^+ 2 == 1) = (`|x| == 1).
+ +
+Lemma lerif_mean_square_scaled x y :
+ x × y *+ 2 ≤ x ^+ 2 + y ^+ 2 ?= iff (x == y).
+ +
+Lemma lerif_AGM2_scaled x y : x × y *+ 4 ≤ (x + y) ^+ 2 ?= iff (x == y).
+ +
+Section MinMax.
+ +
+
+ GG: Many of the first lemmas hold unconditionally, and others hold for
+ the real subset of a general domain.
+
+
+Lemma minrC : @commutative R R min.
+ +
+Lemma minrr : @idempotent R min.
+ +
+Lemma minr_l x y : x ≤ y → min x y = x.
+ +
+Lemma minr_r x y : y ≤ x → min x y = y.
+ +
+Lemma maxrC : @commutative R R max.
+ +
+Lemma maxrr : @idempotent R max.
+ +
+Lemma maxr_l x y : y ≤ x → max x y = x.
+ +
+Lemma maxr_r x y : x ≤ y → max x y = y.
+ +
+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 minrA x y z : min x (min y z) = min (min x y) z.
+ +
+Lemma minrCA : @left_commutative R R min.
+ +
+Lemma minrAC : @right_commutative R R min.
+ +
+CoInductive minr_spec x y : bool → bool → R → Type :=
+| Minr_r of x ≤ y : minr_spec x y true false x
+| Minr_l of y < x : minr_spec x y false true y.
+ +
+Lemma minrP x y : minr_spec x y (x ≤ y) (y < x) (min x y).
+ +
+Lemma oppr_max x y : - max x y = min (- x) (- y).
+ +
+Lemma oppr_min x y : - min x y = max (- x) (- y).
+ +
+Lemma maxrA x y z : max x (max y z) = max (max x y) z.
+ +
+Lemma maxrCA : @left_commutative R R max.
+ +
+Lemma maxrAC : @right_commutative R R max.
+ +
+CoInductive maxr_spec x y : bool → bool → R → Type :=
+| Maxr_r of y ≤ x : maxr_spec x y true false x
+| Maxr_l of x < y : maxr_spec x y false true y.
+ +
+Lemma maxrP x y : maxr_spec x y (y ≤ x) (x < y) (maxr x y).
+ +
+Lemma eqr_minl x y : (min x y == x) = (x ≤ y).
+ +
+Lemma eqr_minr x y : (min x y == y) = (y ≤ x).
+ +
+Lemma eqr_maxl x y : (max x y == x) = (y ≤ x).
+ +
+Lemma eqr_maxr x y : (max x y == y) = (x ≤ y).
+ +
+Lemma ler_minr x y z : (x ≤ min y z) = (x ≤ y) && (x ≤ z).
+ +
+Lemma ler_minl x y z : (min y z ≤ x) = (y ≤ x) || (z ≤ x).
+ +
+Lemma ler_maxr x y z : (x ≤ max y z) = (x ≤ y) || (x ≤ z).
+ +
+Lemma ler_maxl x y z : (max y z ≤ x) = (y ≤ x) && (z ≤ x).
+ +
+Lemma ltr_minr x y z : (x < min y z) = (x < y) && (x < z).
+ +
+Lemma ltr_minl x y z : (min y z < x) = (y < x) || (z < x).
+ +
+Lemma ltr_maxr x y z : (x < max y z) = (x < y) || (x < z).
+ +
+Lemma ltr_maxl x y z : (max y z < x) = (y < x) && (z < x).
+ +
+Definition lter_minr := (ler_minr, ltr_minr).
+Definition lter_minl := (ler_minl, ltr_minl).
+Definition lter_maxr := (ler_maxr, ltr_maxr).
+Definition lter_maxl := (ler_maxl, ltr_maxl).
+ +
+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 minrK x y : max (min x y) x = x.
+ +
+Lemma minKr x y : min y (max x y) = y.
+ +
+Lemma maxr_minl : @left_distributive R R max min.
+ +
+Lemma maxr_minr : @right_distributive R R max min.
+ +
+Lemma minr_maxl : @left_distributive R R min max.
+ +
+Lemma minr_maxr : @right_distributive R R min max.
+ +
+Lemma minr_pmulr x y z : 0 ≤ x → x × min y z = min (x × y) (x × z).
+ +
+Lemma minr_nmulr x y z : x ≤ 0 → x × min y z = max (x × y) (x × z).
+ +
+Lemma maxr_pmulr x y z : 0 ≤ x → x × max y z = max (x × y) (x × z).
+ +
+Lemma maxr_nmulr x y z : x ≤ 0 → x × max y z = min (x × y) (x × z).
+ +
+Lemma minr_pmull x y z : 0 ≤ x → min y z × x = min (y × x) (z × x).
+ +
+Lemma minr_nmull x y z : x ≤ 0 → min y z × x = max (y × x) (z × x).
+ +
+Lemma maxr_pmull x y z : 0 ≤ x → max y z × x = max (y × x) (z × x).
+ +
+Lemma maxr_nmull 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 lerif_mean_square : x × y ≤ (x ^+ 2 + y ^+ 2) / 2%:R ?= iff (x == y).
+ +
+Lemma lerif_AGM2 : x × y ≤ ((x + y) / 2%:R)^+ 2 ?= iff (x == y).
+ +
+End RealField.
+ +
+Section ArchimedeanFieldTheory.
+ +
+Variables (F : archiFieldType) (x : F).
+ +
+Lemma archi_boundP : 0 ≤ x → x < (bound x)%:R.
+ +
+Lemma upper_nthrootP i : (bound x ≤ i)%N → x < 2%:R ^+ i.
+ +
+End ArchimedeanFieldTheory.
+ +
+Section RealClosedFieldTheory.
+ +
+Variable R : rcfType.
+Implicit Types a x y : R.
+ +
+Lemma poly_ivt : real_closed_axiom R.
+ +
+
+
++ +
+Lemma minrr : @idempotent R min.
+ +
+Lemma minr_l x y : x ≤ y → min x y = x.
+ +
+Lemma minr_r x y : y ≤ x → min x y = y.
+ +
+Lemma maxrC : @commutative R R max.
+ +
+Lemma maxrr : @idempotent R max.
+ +
+Lemma maxr_l x y : y ≤ x → max x y = x.
+ +
+Lemma maxr_r x y : x ≤ y → max x y = y.
+ +
+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 minrA x y z : min x (min y z) = min (min x y) z.
+ +
+Lemma minrCA : @left_commutative R R min.
+ +
+Lemma minrAC : @right_commutative R R min.
+ +
+CoInductive minr_spec x y : bool → bool → R → Type :=
+| Minr_r of x ≤ y : minr_spec x y true false x
+| Minr_l of y < x : minr_spec x y false true y.
+ +
+Lemma minrP x y : minr_spec x y (x ≤ y) (y < x) (min x y).
+ +
+Lemma oppr_max x y : - max x y = min (- x) (- y).
+ +
+Lemma oppr_min x y : - min x y = max (- x) (- y).
+ +
+Lemma maxrA x y z : max x (max y z) = max (max x y) z.
+ +
+Lemma maxrCA : @left_commutative R R max.
+ +
+Lemma maxrAC : @right_commutative R R max.
+ +
+CoInductive maxr_spec x y : bool → bool → R → Type :=
+| Maxr_r of y ≤ x : maxr_spec x y true false x
+| Maxr_l of x < y : maxr_spec x y false true y.
+ +
+Lemma maxrP x y : maxr_spec x y (y ≤ x) (x < y) (maxr x y).
+ +
+Lemma eqr_minl x y : (min x y == x) = (x ≤ y).
+ +
+Lemma eqr_minr x y : (min x y == y) = (y ≤ x).
+ +
+Lemma eqr_maxl x y : (max x y == x) = (y ≤ x).
+ +
+Lemma eqr_maxr x y : (max x y == y) = (x ≤ y).
+ +
+Lemma ler_minr x y z : (x ≤ min y z) = (x ≤ y) && (x ≤ z).
+ +
+Lemma ler_minl x y z : (min y z ≤ x) = (y ≤ x) || (z ≤ x).
+ +
+Lemma ler_maxr x y z : (x ≤ max y z) = (x ≤ y) || (x ≤ z).
+ +
+Lemma ler_maxl x y z : (max y z ≤ x) = (y ≤ x) && (z ≤ x).
+ +
+Lemma ltr_minr x y z : (x < min y z) = (x < y) && (x < z).
+ +
+Lemma ltr_minl x y z : (min y z < x) = (y < x) || (z < x).
+ +
+Lemma ltr_maxr x y z : (x < max y z) = (x < y) || (x < z).
+ +
+Lemma ltr_maxl x y z : (max y z < x) = (y < x) && (z < x).
+ +
+Definition lter_minr := (ler_minr, ltr_minr).
+Definition lter_minl := (ler_minl, ltr_minl).
+Definition lter_maxr := (ler_maxr, ltr_maxr).
+Definition lter_maxl := (ler_maxl, ltr_maxl).
+ +
+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 minrK x y : max (min x y) x = x.
+ +
+Lemma minKr x y : min y (max x y) = y.
+ +
+Lemma maxr_minl : @left_distributive R R max min.
+ +
+Lemma maxr_minr : @right_distributive R R max min.
+ +
+Lemma minr_maxl : @left_distributive R R min max.
+ +
+Lemma minr_maxr : @right_distributive R R min max.
+ +
+Lemma minr_pmulr x y z : 0 ≤ x → x × min y z = min (x × y) (x × z).
+ +
+Lemma minr_nmulr x y z : x ≤ 0 → x × min y z = max (x × y) (x × z).
+ +
+Lemma maxr_pmulr x y z : 0 ≤ x → x × max y z = max (x × y) (x × z).
+ +
+Lemma maxr_nmulr x y z : x ≤ 0 → x × max y z = min (x × y) (x × z).
+ +
+Lemma minr_pmull x y z : 0 ≤ x → min y z × x = min (y × x) (z × x).
+ +
+Lemma minr_nmull x y z : x ≤ 0 → min y z × x = max (y × x) (z × x).
+ +
+Lemma maxr_pmull x y z : 0 ≤ x → max y z × x = max (y × x) (z × x).
+ +
+Lemma maxr_nmull 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 lerif_mean_square : x × y ≤ (x ^+ 2 + y ^+ 2) / 2%:R ?= iff (x == y).
+ +
+Lemma lerif_AGM2 : x × y ≤ ((x + y) / 2%:R)^+ 2 ?= iff (x == y).
+ +
+End RealField.
+ +
+Section ArchimedeanFieldTheory.
+ +
+Variables (F : archiFieldType) (x : F).
+ +
+Lemma archi_boundP : 0 ≤ x → x < (bound x)%:R.
+ +
+Lemma upper_nthrootP i : (bound x ≤ i)%N → x < 2%:R ^+ i.
+ +
+End ArchimedeanFieldTheory.
+ +
+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.
+ +
+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.
+ +
+CoInductive 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 @pos 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).
+ +
+End RealClosedFieldTheory.
+ +
+Definition conjC {C : numClosedFieldType} : {rmorphism C → C} :=
+ ClosedField.conj_op (ClosedField.conj_mixin (ClosedField.class C)).
+Notation "z ^*" := (@conjC _ z) (at level 2, format "z ^*") : ring_scope.
+ +
+Definition imaginaryC {C : numClosedFieldType} : C :=
+ ClosedField.imaginary (ClosedField.conj_mixin (ClosedField.class C)).
+Notation "'i" := (@imaginaryC _) (at level 0) : ring_scope.
+ +
+Section ClosedFieldTheory.
+ +
+Variable C : numClosedFieldType.
+Implicit Types a x y z : C.
+ +
+Definition normCK x : `|x| ^+ 2 = x × x^*.
+ +
+Lemma sqrCi : 'i ^+ 2 = -1 :> C.
+ +
+Lemma conjCK : involutive (@conjC C).
+ +
+Let Re2 z := z + z^*.
+Definition nnegIm z := (0 ≤ imaginaryC × (z^* - z)).
+Definition argCle y z := nnegIm z ==> nnegIm y && (Re2 z ≤ Re2 y).
+ +
+CoInductive 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) (at level 2, format "n .-root") : ring_core_scope.
+Notation "n .-root" := (nthroot n) (only parsing) : ring_scope.
+Notation sqrtC := 2.-root.
+ +
+Definition Re x := (x + x^*) / 2%:R.
+Definition Im x := 'i × (x^* - x) / 2%:R.
+Notation "'Re z" := (Re z) (at level 10, z at level 8) : ring_scope.
+Notation "'Im z" := (Im z) (at level 10, z at level 8) : ring_scope.
+ +
+Let nz2 : 2%:R != 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 conjC_eq0 x : (x^* == 0) = (x == 0).
+ +
+Lemma invC_norm x : x^-1 = `|x| ^- 2 × x^*.
+ +
+
+
++Lemma sqrtr_ge0 a : 0 ≤ sqrt a.
+ Hint Resolve sqrtr_ge0.
+ +
+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.
+ +
+CoInductive 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 @pos 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).
+ +
+End RealClosedFieldTheory.
+ +
+Definition conjC {C : numClosedFieldType} : {rmorphism C → C} :=
+ ClosedField.conj_op (ClosedField.conj_mixin (ClosedField.class C)).
+Notation "z ^*" := (@conjC _ z) (at level 2, format "z ^*") : ring_scope.
+ +
+Definition imaginaryC {C : numClosedFieldType} : C :=
+ ClosedField.imaginary (ClosedField.conj_mixin (ClosedField.class C)).
+Notation "'i" := (@imaginaryC _) (at level 0) : ring_scope.
+ +
+Section ClosedFieldTheory.
+ +
+Variable C : numClosedFieldType.
+Implicit Types a x y z : C.
+ +
+Definition normCK x : `|x| ^+ 2 = x × x^*.
+ +
+Lemma sqrCi : 'i ^+ 2 = -1 :> C.
+ +
+Lemma conjCK : involutive (@conjC C).
+ +
+Let Re2 z := z + z^*.
+Definition nnegIm z := (0 ≤ imaginaryC × (z^* - z)).
+Definition argCle y z := nnegIm z ==> nnegIm y && (Re2 z ≤ Re2 y).
+ +
+CoInductive 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) (at level 2, format "n .-root") : ring_core_scope.
+Notation "n .-root" := (nthroot n) (only parsing) : ring_scope.
+Notation sqrtC := 2.-root.
+ +
+Definition Re x := (x + x^*) / 2%:R.
+Definition Im x := 'i × (x^* - x) / 2%:R.
+Notation "'Re z" := (Re z) (at level 10, z at level 8) : ring_scope.
+Notation "'Im z" := (Im z) (at level 10, z at level 8) : ring_scope.
+ +
+Let nz2 : 2%:R != 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 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 geC0_conj x : 0 ≤ x → x^* = x.
+ +
+Lemma geC0_unit_exp x n : 0 ≤ x → (x ^+ n.+1 == 1) = (x == 1).
+ +
+
+
++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 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).
+ +
+
+
++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 Creal_Re x : 'Re x \is real.
+ +
+Lemma Creal_Im x : 'Im x \is real.
+Hint Resolve Creal_Re Creal_Im.
+ +
+Fact Re_is_additive : additive Re.
+ Canonical Re_additive := Additive Re_is_additive.
+ +
+Fact Im_is_additive : additive Im.
+Canonical Im_additive := Additive 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 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_rect :
+ {in real &, ∀ x y, (x + 'i × y)^-1 = (x - 'i × y) / (x ^+ 2 + y ^+ 2)}.
+ +
+Lemma lerif_normC_Re_Creal z : `|'Re z| ≤ `|z| ?= iff (z \is real).
+ +
+Lemma lerif_Re_Creal z : 'Re z ≤ `|z| ?= iff (0 ≤ z).
+ +
+
+
++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 Creal_Re x : 'Re x \is real.
+ +
+Lemma Creal_Im x : 'Im x \is real.
+Hint Resolve Creal_Re Creal_Im.
+ +
+Fact Re_is_additive : additive Re.
+ Canonical Re_additive := Additive Re_is_additive.
+ +
+Fact Im_is_additive : additive Im.
+Canonical Im_additive := Additive 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 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_rect :
+ {in real &, ∀ x y, (x + 'i × y)^-1 = (x - 'i × y) / (x ^+ 2 + y ^+ 2)}.
+ +
+Lemma lerif_normC_Re_Creal z : `|'Re z| ≤ `|z| ?= iff (z \is real).
+ +
+Lemma lerif_Re_Creal z : 'Re z ≤ `|z| ?= iff (0 ≤ z).
+ +
+
+ Equality from polar coordinates, for the upper plane.
+
+
+
+
+ Nth roots.
+
+
+
+
+ case Du: sqrCi => [u u2N1] /=.
+ have/eqP := u2N1; rewrite -sqrCi eqf_sqr => /pred2P[ ] //.
+ have:= conjCi; rewrite /'i; case_rootC => /= v v2n1 min_v conj_v Duv.
+ have{min_v} /idPn[ ] := min_v u isT u2N1; rewrite negb_imply /nnegIm Du /= Duv.
+ rewrite rmorphN conj_v opprK -opprD mulrNN mulNr -mulr2n mulrnAr -expr2 v2n1.
+ by rewrite mulNrn opprK ler0n oppr_ge0 (ler_nat _ 2 0).
+
+
+
+
+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 : (n > 0)%N → 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).
+ +
+
+
++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 : (n > 0)%N → 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 lerif_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].
+ +
+
+
++Lemma lerif_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 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).
+ +
+
+
++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 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 normC_add_eq 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 normC_sub_eq x y :
+ `|x - y| = `|x| - `|y| → {t | `|t| == 1 & (x, y) = (`|x| × t, `|y| × t)}.
+ +
+End ClosedFieldTheory.
+ +
+Notation "n .-root" := (@nthroot _ n) (at level 2, format "n .-root") : ring_scope.
+Notation sqrtC := 2.-root.
+Notation "'i" := (@imaginaryC _) (at level 0) : ring_scope.
+Notation "'Re z" := (Re z) (at level 10, z at level 8) : ring_scope.
+Notation "'Im z" := (Im z) (at level 10, z at level 8) : ring_scope.
+ +
+End Theory.
+ +
+Module RealMixin.
+ +
+Section RealMixins.
+ +
+Variables (R : idomainType) (le : rel R) (lt : rel R) (norm : R → R).
+ +
+Section LeMixin.
+ +
+Hypothesis le0_add : ∀ x y, 0 ≤ x → 0 ≤ y → 0 ≤ x + y.
+Hypothesis le0_mul : ∀ x y, 0 ≤ x → 0 ≤ y → 0 ≤ x × y.
+Hypothesis le0_anti : ∀ x, 0 ≤ x → x ≤ 0 → x = 0.
+Hypothesis sub_ge0 : ∀ x y, (0 ≤ y - x) = (x ≤ y).
+Hypothesis le0_total : ∀ x, (0 ≤ x) || (x ≤ 0).
+Hypothesis normN: ∀ x, `|- x| = `|x|.
+Hypothesis ge0_norm : ∀ x, 0 ≤ x → `|x| = x.
+Hypothesis lt_def : ∀ x y, (x < y) = (y != x) && (x ≤ y).
+ +
+Let le0N x : (0 ≤ - x) = (x ≤ 0).
+Let leN_total x : 0 ≤ x ∨ 0 ≤ - x.
+ +
+Let le00 : (0 ≤ 0).
+Let le01 : (0 ≤ 1).
+ +
+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|.
+ +
+Lemma le_total x y : (x ≤ y) || (y ≤ x).
+ +
+Definition Le :=
+ Mixin le_normD lt0_add eq0_norm (in2W le_total) normM le_def lt_def.
+ +
+Lemma Real (R' : numDomainType) & phant R' :
+ R' = NumDomainType R Le → real_axiom R'.
+ +
+End LeMixin.
+ +
+Section LtMixin.
+ +
+Hypothesis lt0_add : ∀ x y, 0 < x → 0 < y → 0 < x + y.
+Hypothesis lt0_mul : ∀ x y, 0 < x → 0 < y → 0 < x × y.
+Hypothesis lt0_ngt0 : ∀ x, 0 < x → ~~ (x < 0).
+Hypothesis sub_gt0 : ∀ x y, (0 < y - x) = (x < y).
+Hypothesis lt0_total : ∀ x, x != 0 → (0 < x) || (x < 0).
+Hypothesis normN : ∀ x, `|- x| = `|x|.
+Hypothesis ge0_norm : ∀ x, 0 ≤ x → `|x| = x.
+Hypothesis le_def : ∀ x y, (x ≤ y) = (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 le0_anti x : 0 ≤ x → x ≤ 0 → x = 0.
+ +
+Fact sub_ge0 x y : (0 ≤ y - x) = (x ≤ y).
+ +
+Fact lt_def x y : (x < y) = (y != x) && (x ≤ y).
+ +
+Fact le0_total x : (0 ≤ x) || (x ≤ 0).
+ +
+Definition Lt :=
+ Le le0_add le0_mul le0_anti sub_ge0 le0_total normN ge0_norm lt_def.
+ +
+End LtMixin.
+ +
+End RealMixins.
+ +
+End RealMixin.
+ +
+End Num.
+ +
+Export Num.NumDomain.Exports Num.NumField.Exports Num.ClosedField.Exports.
+Export Num.RealDomain.Exports Num.RealField.Exports.
+Export Num.ArchimedeanField.Exports Num.RealClosedField.Exports.
+Export Num.Syntax Num.PredInstances.
+ +
+Notation RealLeMixin := Num.RealMixin.Le.
+Notation RealLtMixin := Num.RealMixin.Lt.
+Notation RealLeAxiom R := (Num.RealMixin.Real (Phant R) (erefl _)).
+Notation ImaginaryMixin := Num.ClosedField.ImaginaryMixin.
+
++Lemma normC_add_eq 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 normC_sub_eq x y :
+ `|x - y| = `|x| - `|y| → {t | `|t| == 1 & (x, y) = (`|x| × t, `|y| × t)}.
+ +
+End ClosedFieldTheory.
+ +
+Notation "n .-root" := (@nthroot _ n) (at level 2, format "n .-root") : ring_scope.
+Notation sqrtC := 2.-root.
+Notation "'i" := (@imaginaryC _) (at level 0) : ring_scope.
+Notation "'Re z" := (Re z) (at level 10, z at level 8) : ring_scope.
+Notation "'Im z" := (Im z) (at level 10, z at level 8) : ring_scope.
+ +
+End Theory.
+ +
+Module RealMixin.
+ +
+Section RealMixins.
+ +
+Variables (R : idomainType) (le : rel R) (lt : rel R) (norm : R → R).
+ +
+Section LeMixin.
+ +
+Hypothesis le0_add : ∀ x y, 0 ≤ x → 0 ≤ y → 0 ≤ x + y.
+Hypothesis le0_mul : ∀ x y, 0 ≤ x → 0 ≤ y → 0 ≤ x × y.
+Hypothesis le0_anti : ∀ x, 0 ≤ x → x ≤ 0 → x = 0.
+Hypothesis sub_ge0 : ∀ x y, (0 ≤ y - x) = (x ≤ y).
+Hypothesis le0_total : ∀ x, (0 ≤ x) || (x ≤ 0).
+Hypothesis normN: ∀ x, `|- x| = `|x|.
+Hypothesis ge0_norm : ∀ x, 0 ≤ x → `|x| = x.
+Hypothesis lt_def : ∀ x y, (x < y) = (y != x) && (x ≤ y).
+ +
+Let le0N x : (0 ≤ - x) = (x ≤ 0).
+Let leN_total x : 0 ≤ x ∨ 0 ≤ - x.
+ +
+Let le00 : (0 ≤ 0).
+Let le01 : (0 ≤ 1).
+ +
+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|.
+ +
+Lemma le_total x y : (x ≤ y) || (y ≤ x).
+ +
+Definition Le :=
+ Mixin le_normD lt0_add eq0_norm (in2W le_total) normM le_def lt_def.
+ +
+Lemma Real (R' : numDomainType) & phant R' :
+ R' = NumDomainType R Le → real_axiom R'.
+ +
+End LeMixin.
+ +
+Section LtMixin.
+ +
+Hypothesis lt0_add : ∀ x y, 0 < x → 0 < y → 0 < x + y.
+Hypothesis lt0_mul : ∀ x y, 0 < x → 0 < y → 0 < x × y.
+Hypothesis lt0_ngt0 : ∀ x, 0 < x → ~~ (x < 0).
+Hypothesis sub_gt0 : ∀ x y, (0 < y - x) = (x < y).
+Hypothesis lt0_total : ∀ x, x != 0 → (0 < x) || (x < 0).
+Hypothesis normN : ∀ x, `|- x| = `|x|.
+Hypothesis ge0_norm : ∀ x, 0 ≤ x → `|x| = x.
+Hypothesis le_def : ∀ x y, (x ≤ y) = (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 le0_anti x : 0 ≤ x → x ≤ 0 → x = 0.
+ +
+Fact sub_ge0 x y : (0 ≤ y - x) = (x ≤ y).
+ +
+Fact lt_def x y : (x < y) = (y != x) && (x ≤ y).
+ +
+Fact le0_total x : (0 ≤ x) || (x ≤ 0).
+ +
+Definition Lt :=
+ Le le0_add le0_mul le0_anti sub_ge0 le0_total normN ge0_norm lt_def.
+ +
+End LtMixin.
+ +
+End RealMixins.
+ +
+End RealMixin.
+ +
+End Num.
+ +
+Export Num.NumDomain.Exports Num.NumField.Exports Num.ClosedField.Exports.
+Export Num.RealDomain.Exports Num.RealField.Exports.
+Export Num.ArchimedeanField.Exports Num.RealClosedField.Exports.
+Export Num.Syntax Num.PredInstances.
+ +
+Notation RealLeMixin := Num.RealMixin.Le.
+Notation RealLtMixin := Num.RealMixin.Lt.
+Notation RealLeAxiom R := (Num.RealMixin.Real (Phant R) (erefl _)).
+Notation ImaginaryMixin := Num.ClosedField.ImaginaryMixin.
+