(* mathcomp analysis (c) 2017 Inria and AIST. License: CeCILL-C. *) (* -------------------------------------------------------------------- *) (* Copyright (c) - 2015--2016 - IMDEA Software Institute *) (* Copyright (c) - 2015--2018 - Inria *) (* Copyright (c) - 2016--2018 - Polytechnique *) (* -------------------------------------------------------------------- *) From mathcomp Require Import all_ssreflect all_algebra finmap. Require Import boolp mathcomp_extra classical_sets functions reals signed. Require Import topology. (******************************************************************************) (* Extended real numbers *) (* *) (* Given a type R for numbers, \bar R is the type R extended with symbols -oo *) (* and +oo (notation scope: %E), suitable to represent extended real numbers. *) (* When R is a numDomainType, \bar R is equipped with a canonical porderType *) (* and operations for addition/opposite. When R is a realDomainType, \bar R *) (* is equipped with a Canonical orderType. *) (* *) (* Naming convention: in definition/lemma identifiers, "e" stands for an *) (* extended number and "y" and "Ny" for +oo ad -oo respectively. *) (* *) (* \bar R == coproduct of R and {+oo, -oo}; *) (* notation for extended (R:Type) *) (* r%:E == injects real numbers into \bar R *) (* +%E, -%E, *%E == addition/opposite/multiplication for extended *) (* reals *) (* `| x |%E == the absolute value of x *) (* x ^+ n == iterated multiplication *) (* x *+ n == iterated addition *) (* +%dE, (x *+ n)%dE == dual addition/dual iterated addition for *) (* extended reals (-oo + +oo = +oo instead of -oo) *) (* Import DualAddTheory for related lemmas *) (* x +? y == the addition of the extended real numbers x and *) (* and y is defined, i.e., it is neither +oo - oo *) (* nor -oo + oo *) (* x *? y == the multiplication of the extended real numbers *) (* x and y is not of the form 0 * +oo or 0 * -oo *) (* (_ <= _)%E, (_ < _)%E, == comparison relations for extended reals *) (* (_ >= _)%E, (_ > _)%E *) (* (\sum_(i in A) f i)%E == bigop-like notation in scope %E *) (* maxe x y, mine x y == notation for the maximum/minimum of two *) (* extended real numbers *) (* ereal_sup E == supremum of E *) (* ereal_inf E == infimum of E *) (* ereal_supremums_neq0 S == S has a supremum *) (* *) (* Signed extended real numbers: *) (* {posnum \bar R} == interface type for elements in \bar R that are *) (* positive, c.f., signed.v, notation in scope %E *) (* {nonneg \bar R} == interface types for elements in \bar R that are *) (* non-negative, c.f. signed.v, notation in scope %E *) (* x%:pos == explicitly casts x to {posnum \bar R}, in scope %E *) (* x%:nng == explicitly casts x to {nonneg \bar R}, in scope %E *) (* *) (* Topology of extended real numbers: *) (* ereal_topologicalType R == topology for extended real numbers over *) (* R, a realFieldType *) (* contract == order-preserving bijective function *) (* from extended real numbers to [-1; 1] *) (* expand == function from real numbers to extended *) (* real numbers that cancels contract in *) (* [-1; 1] *) (* ereal_pseudoMetricType R == pseudometric space for extended reals *) (* over R where is a realFieldType; the *) (* distance between x and y is defined by *) (* `|contract x - contract y| *) (* *) (* Filters: *) (* ereal_dnbhs x == filter on extended real numbers that *) (* corresponds to the deleted neighborhood *) (* x^' if x is a real number and to *) (* predicates that are eventually true if x *) (* is +oo/-oo. *) (* ereal_nbhs x == same as ereal_dnbhs where dnbhs is *) (* replaced with nbhs. *) (* ereal_loc_seq x == sequence that converges to x in the set *) (* of extended real numbers. *) (* *) (******************************************************************************) Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. Reserved Notation "x %:E" (at level 2, format "x %:E"). Reserved Notation "x +? y" (at level 50, format "x +? y"). Reserved Notation "x *? y" (at level 50, format "x *? y"). Reserved Notation "'\bar' x" (at level 2, format "'\bar' x"). Reserved Notation "{ 'posnum' '\bar' R }" (at level 0, format "{ 'posnum' '\bar' R }"). Reserved Notation "{ 'nonneg' '\bar' R }" (at level 0, format "{ 'nonneg' '\bar' R }"). Declare Scope ereal_dual_scope. Declare Scope ereal_scope. Import Order.TTheory GRing.Theory Num.Theory. Import numFieldTopology.Exports. Local Open Scope ring_scope. Variant extended (R : Type) := EFin of R | EPInf | ENInf. Arguments EFin {R}. Lemma EFin_inj T : injective (@EFin T). Proof. by move=> a b; case. Qed. Definition dual_extended := extended. (* Notations in ereal_dual_scope should be kept *before* the corresponding notation in ereal_scope, otherwise when none of the scope is open (lte x y) would be displayed as (x < y)%dE, instead of (x < y)%E, for instance. *) Notation "+oo" := (@EPInf _ : dual_extended _) : ereal_dual_scope. Notation "+oo" := (@EPInf _) : ereal_scope. Notation "-oo" := (@ENInf _ : dual_extended _) : ereal_dual_scope. Notation "-oo" := (@ENInf _) : ereal_scope. Notation "r %:E" := (@EFin _ r%R). Notation "'\bar' R" := (extended R) : type_scope. Notation "0" := (0%R%:E : dual_extended _) : ereal_dual_scope. Notation "0" := (0%R%:E) : ereal_scope. Notation "1" := (1%R%:E : dual_extended _) : ereal_dual_scope. Notation "1" := (1%R%:E) : ereal_scope. Bind Scope ereal_dual_scope with dual_extended. Bind Scope ereal_scope with extended. Delimit Scope ereal_dual_scope with dE. Delimit Scope ereal_scope with E. Local Open Scope ereal_scope. Local Open Scope classical_set_scope. Lemma EFin_bigcup T (F : nat -> set T) : EFin @` (\bigcup_i F i) = \bigcup_i (EFin @` F i). Proof. rewrite eqEsubset; split => [_ [r [n _ Fnr <-]]|]; first by exists n => //; exists r. by move=> x [n _ [r Fnr <- /=]]; exists r => //; exists n. Qed. Lemma EFin_setC T (A : set T) : EFin @` (~` A) = (~` (EFin @` A)) `\` [set -oo; +oo]. Proof. rewrite eqEsubset; split => [_ [r Ar <-]|[r | |]]. by split => [|[]//]; apply: contra_not Ar => -[? ? [] <-]. - move=> [Ar _]; apply/not_exists2P; apply: contra_not Ar => h. by exists r => //; have [|//] := h r; apply: contrapT. - by move=> -[_] /not_orP[_ /=]. - by move=> -[_] /not_orP[/=]. Qed. Local Close Scope classical_set_scope. Definition er_map T T' (f : T -> T') (x : \bar T) : \bar T' := match x with | r%:E => (f r)%:E | +oo => +oo | -oo => -oo end. Definition fine {R : zmodType} x : R := if x is EFin v then v else 0. Section EqEReal. Variable (R : eqType). Definition eq_ereal (x y : \bar R) := match x, y with | x%:E, y%:E => x == y | +oo, +oo => true | -oo, -oo => true | _, _ => false end. Lemma ereal_eqP : Equality.axiom eq_ereal. Proof. by case=> [?||][?||]; apply: (iffP idP) => //= [/eqP|[]] ->. Qed. Definition ereal_eqMixin := Equality.Mixin ereal_eqP. Canonical ereal_eqType := Equality.Pack ereal_eqMixin. Lemma eqe (r1 r2 : R) : (r1%:E == r2%:E) = (r1 == r2). Proof. by []. Qed. End EqEReal. Section ERealChoice. Variable (R : choiceType). Definition code (x : \bar R) := match x with | r%:E => GenTree.Node 0 [:: GenTree.Leaf r] | +oo => GenTree.Node 1 [::] | -oo => GenTree.Node 2 [::] end. Definition decode (x : GenTree.tree R) : option (\bar R) := match x with | GenTree.Node 0 [:: GenTree.Leaf r] => Some r%:E | GenTree.Node 1 [::] => Some +oo | GenTree.Node 2 [::] => Some -oo | _ => None end. Lemma codeK : pcancel code decode. Proof. by case. Qed. Definition ereal_choiceMixin := PcanChoiceMixin codeK. Canonical ereal_choiceType := ChoiceType (extended R) ereal_choiceMixin. End ERealChoice. Section ERealCount. Variable (R : countType). Definition ereal_countMixin := PcanCountMixin (@codeK R). Canonical ereal_countType := CountType (extended R) ereal_countMixin. End ERealCount. Section ERealOrder. Context {R : numDomainType}. Implicit Types x y : \bar R. Definition le_ereal x1 x2 := match x1, x2 with | -oo, r%:E | r%:E, +oo => r \is Num.real | r1%:E, r2%:E => r1 <= r2 | -oo, _ | _, +oo => true | +oo, _ | _, -oo => false end. Definition lt_ereal x1 x2 := match x1, x2 with | -oo, r%:E | r%:E, +oo => r \is Num.real | r1%:E, r2%:E => r1 < r2 | -oo, -oo | +oo, +oo => false | +oo, _ | _ , -oo => false | -oo, _ => true end. Lemma lt_def_ereal x y : lt_ereal x y = (y != x) && le_ereal x y. Proof. by case: x y => [?||][?||] //=; rewrite lt_def eqe. Qed. Lemma le_refl_ereal : reflexive le_ereal. Proof. by case => /=. Qed. Lemma le_anti_ereal : ssrbool.antisymmetric le_ereal. Proof. by case=> [?||][?||]/=; rewrite ?andbF => // /le_anti ->. Qed. Lemma le_trans_ereal : ssrbool.transitive le_ereal. Proof. case=> [?||][?||][?||] //=; rewrite -?comparabler0; first exact: le_trans. by move=> /le_comparable cmp /(comparabler_trans cmp). by move=> cmp /ge_comparable /comparabler_trans; apply. Qed. Fact ereal_display : unit. Proof. by []. Qed. Definition ereal_porderMixin := LePOrderMixin lt_def_ereal le_refl_ereal le_anti_ereal le_trans_ereal. Canonical ereal_porderType := POrderType ereal_display (extended R) ereal_porderMixin. Lemma leEereal x y : (x <= y)%O = le_ereal x y. Proof. by []. Qed. Lemma ltEereal x y : (x < y)%O = lt_ereal x y. Proof. by []. Qed. End ERealOrder. Notation lee := (@Order.le ereal_display _) (only parsing). Notation "@ 'lee' R" := (@Order.le ereal_display R) (at level 10, R at level 8, only parsing). Notation lte := (@Order.lt ereal_display _) (only parsing). Notation "@ 'lte' R" := (@Order.lt ereal_display R) (at level 10, R at level 8, only parsing). Notation gee := (@Order.ge ereal_display _) (only parsing). Notation "@ 'gee' R" := (@Order.ge ereal_display R) (at level 10, R at level 8, only parsing). Notation gte := (@Order.gt ereal_display _) (only parsing). Notation "@ 'gte' R" := (@Order.gt ereal_display R) (at level 10, R at level 8, only parsing). Notation "x <= y" := (lee x y) (only printing) : ereal_dual_scope. Notation "x <= y" := (lee x y) (only printing) : ereal_scope. Notation "x < y" := (lte x y) (only printing) : ereal_dual_scope. Notation "x < y" := (lte x y) (only printing) : ereal_scope. Notation "x <= y <= z" := ((lee x y) && (lee y z)) (only printing) : ereal_dual_scope. Notation "x <= y <= z" := ((lee x y) && (lee y z)) (only printing) : ereal_scope. Notation "x < y <= z" := ((lte x y) && (lee y z)) (only printing) : ereal_dual_scope. Notation "x < y <= z" := ((lte x y) && (lee y z)) (only printing) : ereal_scope. Notation "x <= y < z" := ((lee x y) && (lte y z)) (only printing) : ereal_dual_scope. Notation "x <= y < z" := ((lee x y) && (lte y z)) (only printing) : ereal_scope. Notation "x < y < z" := ((lte x y) && (lte y z)) (only printing) : ereal_dual_scope. Notation "x < y < z" := ((lte x y) && (lte y z)) (only printing) : ereal_scope. Notation "x <= y" := (lee (x%dE : dual_extended _) (y%dE : dual_extended _)) : ereal_dual_scope. Notation "x <= y" := (lee (x : extended _) (y : extended _)) : ereal_scope. Notation "x < y" := (lte (x%dE : dual_extended _) (y%dE : dual_extended _)) : ereal_dual_scope. Notation "x < y" := (lte (x : extended _) (y : extended _)) : ereal_scope. Notation "x >= y" := (y <= x) (only parsing) : ereal_dual_scope. Notation "x >= y" := (y <= x) (only parsing) : ereal_scope. Notation "x > y" := (y < x) (only parsing) : ereal_dual_scope. Notation "x > y" := (y < x) (only parsing) : ereal_scope. Notation "x <= y <= z" := ((x <= y) && (y <= z)) : ereal_dual_scope. Notation "x <= y <= z" := ((x <= y) && (y <= z)) : ereal_scope. Notation "x < y <= z" := ((x < y) && (y <= z)) : ereal_dual_scope. Notation "x < y <= z" := ((x < y) && (y <= z)) : ereal_scope. Notation "x <= y < z" := ((x <= y) && (y < z)) : ereal_dual_scope. Notation "x <= y < z" := ((x <= y) && (y < z)) : ereal_scope. Notation "x < y < z" := ((x < y) && (y < z)) : ereal_dual_scope. Notation "x < y < z" := ((x < y) && (y < z)) : ereal_scope. Notation "x <= y :> T" := ((x : T) <= (y : T)) (only parsing) : ereal_scope. Notation "x < y :> T" := ((x : T) < (y : T)) (only parsing) : ereal_scope. Section ERealOrder_numDomainType. Context {R : numDomainType}. Implicit Types x y : \bar R. Lemma lee_fin (r s : R) : (r%:E <= s%:E) = (r <= s)%R. Proof. by []. Qed. Lemma lte_fin (r s : R) : (r%:E < s%:E) = (r < s)%R. Proof. by []. Qed. Lemma lee01 : 0 <= 1 :> \bar R. Proof. by rewrite lee_fin. Qed. Lemma lte01 : 0 < 1 :> \bar R. Proof. by rewrite lte_fin. Qed. Lemma leeNy_eq x : (x <= -oo) = (x == -oo). Proof. by case: x. Qed. Lemma leye_eq x : (+oo <= x) = (x == +oo). Proof. by case: x. Qed. Lemma lt0y : (0 : \bar R) < +oo. Proof. exact: real0. Qed. Lemma ltNy0 : -oo < (0 : \bar R). Proof. exact: real0. Qed. Lemma le0y : (0 : \bar R) <= +oo. Proof. exact: real0. Qed. Lemma leNy0 : -oo <= (0 : \bar R). Proof. exact: real0. Qed. Lemma ereal_comparable x y : (0%E >=< x)%O -> (0%E >=< y)%O -> (x >=< y)%O. Proof. move: x y => [x||] [y||] //; rewrite /Order.comparable !lee_fin -!realE. - exact: real_comparable. - by rewrite /lee/= => ->. - by rewrite /lee/= => _ ->. Qed. End ERealOrder_numDomainType. #[global] Hint Resolve lee01 lte01 : core. Section ERealOrder_realDomainType. Context {R : realDomainType}. Implicit Types (x y : \bar R) (r : R). Lemma ltey r : r%:E < +oo. Proof. exact: num_real. Qed. Lemma ltNye r : -oo < r%:E. Proof. exact: num_real. Qed. Lemma leey x : x <= +oo. Proof. by case: x => //= r; exact: num_real. Qed. Lemma leNye x : -oo <= x. Proof. by case: x => //= r; exact: num_real. Qed. Lemma gee0P x : 0 <= x <-> x = +oo \/ exists2 r, (r >= 0)%R & x = r%:E. Proof. split=> [|[->|[r r0 ->//]]]; last exact: leey. by case: x => [r r0 | _ |//]; [right; exists r|left]. Qed. Lemma le_total_ereal : totalPOrderMixin [porderType of \bar R]. Proof. by move=> [?||][?||]//=; rewrite (ltEereal, leEereal)/= ?num_real ?le_total. Qed. Canonical ereal_latticeType := LatticeType (extended R) le_total_ereal. Canonical ereal_distrLatticeType := DistrLatticeType (extended R) le_total_ereal. Canonical ereal_orderType := OrderType (extended R) le_total_ereal. End ERealOrder_realDomainType. Section ERealArith. Context {R : numDomainType}. Implicit Types x y z : \bar R. Definition adde_subdef x y := match x, y with | x%:E , y%:E => (x + y)%:E | -oo, _ => -oo | _ , -oo => -oo | +oo, _ => +oo | _ , +oo => +oo end. Definition adde := nosimpl adde_subdef. Definition dual_adde_subdef x y := match x, y with | x%:E , y%:E => (x + y)%R%:E | +oo, _ => +oo | _ , +oo => +oo | -oo, _ => -oo | _ , -oo => -oo end. Definition dual_adde := nosimpl dual_adde_subdef. Definition oppe x := match x with | r%:E => (- r)%:E | -oo => +oo | +oo => -oo end. Definition mule_subdef x y := match x, y with | x%:E , y%:E => (x * y)%:E | -oo, y | y, -oo => if y == 0 then 0 else if 0 < y then -oo else +oo | +oo, y | y, +oo => if y == 0 then 0 else if 0 < y then +oo else -oo end. Definition mule := nosimpl mule_subdef. Definition abse x := if x is r%:E then `|r|%:E else +oo. Definition expe x n := iterop n mule x 1. Definition enatmul x n := iterop n adde x 0. Definition ednatmul x n := iterop n dual_adde x 0. Local Open Scope classical_set_scope. Lemma preimage_abse_pinfty : abse @^-1` [set +oo] = [set -oo; +oo]. Proof. by rewrite predeqE => y; split ; move: y => [y//| |]//=; [right | left | case]. Qed. Local Close Scope classical_set_scope. End ERealArith. Notation "+%dE" := dual_adde. Notation "+%E" := adde. Notation "-%E" := oppe. Notation "x + y" := (dual_adde x%dE y%dE) : ereal_dual_scope. Notation "x + y" := (adde x y) : ereal_scope. Notation "x - y" := (dual_adde x%dE (oppe y%dE)) : ereal_dual_scope. Notation "x - y" := (adde x (oppe y)) : ereal_scope. Notation "- x" := (oppe (x%dE : dual_extended _)) : ereal_dual_scope. Notation "- x" := (oppe x) : ereal_scope. Notation "*%E" := mule. Notation "x * y" := (mule (x%dE : dual_extended _) (y%dE : dual_extended _)) : ereal_dual_scope. Notation "x * y" := (mule x y) : ereal_scope. Notation "`| x |" := (abse (x%dE : dual_extended _)) : ereal_dual_scope. Notation "`| x |" := (abse x) : ereal_scope. Arguments abse {R}. Notation "x ^+ n" := (expe x%dE n) : ereal_dual_scope. Notation "x ^+ n" := (expe x n) : ereal_scope. Notation "x *+ n" := (ednatmul x%dE n) : ereal_dual_scope. Notation "x *+ n" := (enatmul x n) : ereal_scope. Notation "\- f" := (fun x => - f x)%dE : ereal_dual_scope. Notation "\- f" := (fun x => - f x)%E : ereal_scope. Notation "f \+ g" := (fun x => f x + g x)%dE : ereal_dual_scope. Notation "f \+ g" := (fun x => f x + g x)%E : ereal_scope. Notation "f \* g" := (fun x => f x * g x)%dE : ereal_dual_scope. Notation "f \* g" := (fun x => f x * g x)%E : ereal_scope. Notation "f \- g" := (fun x => f x - g x)%dE : ereal_dual_scope. Notation "f \- g" := (fun x => f x - g x)%E : ereal_scope. Notation "\sum_ ( i <- r | P ) F" := (\big[+%dE/0%:E]_(i <- r | P%B) F%dE) : ereal_dual_scope. Notation "\sum_ ( i <- r | P ) F" := (\big[+%E/0%:E]_(i <- r | P%B) F%E) : ereal_scope. Notation "\sum_ ( i <- r ) F" := (\big[+%dE/0%:E]_(i <- r) F%dE) : ereal_dual_scope. Notation "\sum_ ( i <- r ) F" := (\big[+%E/0%:E]_(i <- r) F%E) : ereal_scope. Notation "\sum_ ( m <= i < n | P ) F" := (\big[+%dE/0%:E]_(m <= i < n | P%B) F%dE) : ereal_dual_scope. Notation "\sum_ ( m <= i < n | P ) F" := (\big[+%E/0%:E]_(m <= i < n | P%B) F%E) : ereal_scope. Notation "\sum_ ( m <= i < n ) F" := (\big[+%dE/0%:E]_(m <= i < n) F%dE) : ereal_dual_scope. Notation "\sum_ ( m <= i < n ) F" := (\big[+%E/0%:E]_(m <= i < n) F%E) : ereal_scope. Notation "\sum_ ( i | P ) F" := (\big[+%dE/0%:E]_(i | P%B) F%dE) : ereal_dual_scope. Notation "\sum_ ( i | P ) F" := (\big[+%E/0%:E]_(i | P%B) F%E) : ereal_scope. Notation "\sum_ i F" := (\big[+%dE/0%:E]_i F%dE) : ereal_dual_scope. Notation "\sum_ i F" := (\big[+%E/0%:E]_i F%E) : ereal_scope. Notation "\sum_ ( i : t | P ) F" := (\big[+%dE/0%:E]_(i : t | P%B) F%dE) (only parsing) : ereal_dual_scope. Notation "\sum_ ( i : t | P ) F" := (\big[+%E/0%:E]_(i : t | P%B) F%E) (only parsing) : ereal_scope. Notation "\sum_ ( i : t ) F" := (\big[+%dE/0%:E]_(i : t) F%dE) (only parsing) : ereal_dual_scope. Notation "\sum_ ( i : t ) F" := (\big[+%E/0%:E]_(i : t) F%E) (only parsing) : ereal_scope. Notation "\sum_ ( i < n | P ) F" := (\big[+%dE/0%:E]_(i < n | P%B) F%dE) : ereal_dual_scope. Notation "\sum_ ( i < n | P ) F" := (\big[+%E/0%:E]_(i < n | P%B) F%E) : ereal_scope. Notation "\sum_ ( i < n ) F" := (\big[+%dE/0%:E]_(i < n) F%dE) : ereal_dual_scope. Notation "\sum_ ( i < n ) F" := (\big[+%E/0%:E]_(i < n) F%E) : ereal_scope. Notation "\sum_ ( i 'in' A | P ) F" := (\big[+%dE/0%:E]_(i in A | P%B) F%dE) : ereal_dual_scope. Notation "\sum_ ( i 'in' A | P ) F" := (\big[+%E/0%:E]_(i in A | P%B) F%E) : ereal_scope. Notation "\sum_ ( i 'in' A ) F" := (\big[+%dE/0%:E]_(i in A) F%dE) : ereal_dual_scope. Notation "\sum_ ( i 'in' A ) F" := (\big[+%E/0%:E]_(i in A) F%E) : ereal_scope. Section ERealOrderTheory. Context {R : numDomainType}. Implicit Types x y z : \bar R. Local Tactic Notation "elift" constr(lm) ":" ident(x) := by case: x => [||?]; first by rewrite ?eqe; apply: lm. Local Tactic Notation "elift" constr(lm) ":" ident(x) ident(y) := by case: x y => [?||] [?||]; first by rewrite ?eqe; apply: lm. Local Tactic Notation "elift" constr(lm) ":" ident(x) ident(y) ident(z) := by case: x y z => [?||] [?||] [?||]; first by rewrite ?eqe; apply: lm. Lemma lee0N1 : 0 <= (-1)%:E :> \bar R = false. Proof. by rewrite lee_fin ler0N1. Qed. Lemma lte0N1 : 0 < (-1)%:E :> \bar R = false. Proof. by rewrite lte_fin ltr0N1. Qed. Lemma le0R x : 0 <= x -> (0 <= fine x)%R. Proof. by case: x. Qed. Lemma lt0R x : 0 < x < +oo -> (0 < fine x)%R. Proof. by move: x => [x| |] //=; rewrite ?ltxx ?andbF// lte_fin => /andP[]. Qed. Lemma lee_tofin (r0 r1 : R) : (r0 <= r1)%R -> r0%:E <= r1%:E. Proof. by []. Qed. Lemma lte_tofin (r0 r1 : R) : (r0 < r1)%R -> r0%:E < r1%:E. Proof. by []. Qed. Lemma enatmul_pinfty n : +oo *+ n.+1 = +oo :> \bar R. Proof. by elim: n => //= n ->. Qed. Lemma enatmul_ninfty n : -oo *+ n.+1 = -oo :> \bar R. Proof. by elim: n => //= n ->. Qed. Lemma EFin_natmul (r : R) n : (r *+ n.+1)%:E = r%:E *+ n.+1. Proof. by elim: n => //= n <-. Qed. Lemma mule2n x : x *+ 2 = x + x. Proof. by []. Qed. Lemma expe2 x : x ^+ 2 = x * x. Proof. by []. Qed. End ERealOrderTheory. Section finNumPred. Context {R : numDomainType}. Implicit Type (x : \bar R). Definition fin_num := [qualify a x : \bar R | (x != -oo) && (x != +oo)]. Fact fin_num_key : pred_key fin_num. by []. Qed. Canonical fin_num_keyd := KeyedQualifier fin_num_key. Lemma fin_numE x : (x \is a fin_num) = (x != -oo) && (x != +oo). Proof. by []. Qed. Lemma fin_numP x : reflect ((x != -oo) /\ (x != +oo)) (x \is a fin_num). Proof. by apply/(iffP idP) => [/andP//|/andP]. Qed. Lemma fin_numEn x : (x \isn't a fin_num) = (x == -oo) || (x == +oo). Proof. by rewrite fin_numE negb_and ?negbK. Qed. Lemma fin_numPn x : reflect (x = -oo \/ x = +oo) (x \isn't a fin_num). Proof. by rewrite fin_numEn; apply: (iffP orP) => -[]/eqP; by [left|right]. Qed. End finNumPred. Section ERealArithTh_numDomainType. Context {R : numDomainType}. Implicit Types (x y z : \bar R) (r : R). Lemma fine_eq0 x : x \is a fin_num -> (fine x == 0%R) = (x == 0). Proof. by move: x => [//| |] /=; rewrite fin_numE. Qed. Lemma EFinN r : (- r)%:E = (- r%:E). Proof. by []. Qed. Lemma fineN x : fine (- x) = (- fine x)%R. Proof. by case: x => //=; rewrite oppr0. Qed. Lemma EFinD r r' : (r + r')%:E = r%:E + r'%:E. Proof. by []. Qed. Lemma EFinB r r' : (r - r')%:E = r%:E - r'%:E. Proof. by []. Qed. Lemma EFinM r r' : (r * r')%:E = r%:E * r'%:E. Proof. by []. Qed. Lemma sumEFin I s P (F : I -> R) : \sum_(i <- s | P i) (F i)%:E = (\sum_(i <- s | P i) F i)%:E. Proof. by rewrite (big_morph _ EFinD erefl). Qed. Definition adde_def x y := ~~ ((x == +oo) && (y == -oo)) && ~~ ((x == -oo) && (y == +oo)). Local Notation "x +? y" := (adde_def x y). Lemma adde_defC x y : x +? y = y +? x. Proof. by rewrite /adde_def andbC (andbC (x == -oo)) (andbC (x == +oo)). Qed. Lemma adde_defNN x y : - x +? - y = x +? y. Proof. by move: x y => [x| |] [y| |]. Qed. Lemma adde_defEninfty x : (x +? -oo) = (x != +oo). Proof. by case: x. Qed. Lemma ge0_adde_def : {in [pred x | x >= 0] &, forall x y, x +? y}. Proof. by move=> [x| |] [y| |]. Qed. Lemma addeC : commutative (S := \bar R) +%E. Proof. by case=> [x||] [y||] //; rewrite /adde /= addrC. Qed. Lemma adde0 : right_id (0 : \bar R) +%E. Proof. by case=> // r; rewrite /adde /= addr0. Qed. Lemma add0e : left_id (0 : \bar R) +%E. Proof. by move=> x; rewrite addeC adde0. Qed. Lemma addeA : associative (S := \bar R) +%E. Proof. by case=> [x||] [y||] [z||] //; rewrite /adde /= addrA. Qed. Canonical adde_monoid := Monoid.Law addeA add0e adde0. Canonical adde_comoid := Monoid.ComLaw addeC. Lemma addeAC : @right_commutative (\bar R) _ +%E. Proof. exact: Monoid.mulmAC. Qed. Lemma addeCA : @left_commutative (\bar R) _ +%E. Proof. exact: Monoid.mulmCA. Qed. Lemma addeACA : @interchange (\bar R) +%E +%E. Proof. exact: Monoid.mulmACA. Qed. Lemma adde_gt0 x y : 0 < x -> 0 < y -> 0 < x + y. Proof. by move: x y => [x| |] [y| |] //; rewrite !lte_fin; exact: addr_gt0. Qed. Lemma padde_eq0 x y : 0 <= x -> 0 <= y -> (x + y == 0) = (x == 0) && (y == 0). Proof. move: x y => [x| |] [y| |]//; rewrite !lee_fin; first exact: paddr_eq0. by move=> x0 _ /=; rewrite andbF. Qed. Lemma nadde_eq0 x y : x <= 0 -> y <= 0 -> (x + y == 0) = (x == 0) && (y == 0). Proof. move: x y => [x| |] [y| |]//; rewrite !lee_fin; first exact: naddr_eq0. by move=> x0 _ /=; rewrite andbF. Qed. Lemma realDe x y : (0%E >=< x)%O -> (0%E >=< y)%O -> (0%E >=< x + y)%O. Proof. case: x y => [x||] [y||] //; exact: realD. Qed. Lemma oppe0 : - 0 = 0 :> \bar R. Proof. by rewrite /= oppr0. Qed. Lemma oppeK : involutive (A := \bar R) -%E. Proof. by case=> [x||] //=; rewrite opprK. Qed. Lemma oppe_eq0 x : (- x == 0)%E = (x == 0)%E. Proof. by rewrite -(can_eq oppeK) oppe0. Qed. Lemma oppeD x y : y \is a fin_num -> - (x + y) = - x - y. Proof. by move: x y => [x| |] [y| |] //= _; rewrite opprD. Qed. Lemma sube0 x : x - 0 = x. Proof. by move: x => [x| |] //; rewrite -EFinB subr0. Qed. Lemma sub0e x : 0 - x = - x. Proof. by move: x => [x| |] //; rewrite -EFinB sub0r. Qed. Lemma muleC x y : x * y = y * x. Proof. by move: x y => [r||] [s||]//=; rewrite -EFinM mulrC. Qed. Lemma onee_neq0 : 1 != 0 :> \bar R. Proof. exact: oner_neq0. Qed. Lemma onee_eq0 : 1 == 0 :> \bar R = false. Proof. exact: oner_eq0. Qed. Lemma mule1 x : x * 1 = x. Proof. by move: x=> [r||]/=; rewrite /mule/= ?mulr1 ?(negbTE onee_neq0) ?lte_tofin. Qed. Lemma mul1e x : 1 * x = x. Proof. by rewrite muleC mule1. Qed. Lemma mule0 x : x * 0 = 0. Proof. by move: x => [r| |] //=; rewrite /mule/= ?mulr0// eqxx. Qed. Lemma mul0e x : 0 * x = 0. Proof. by move: x => [r| |]/=; rewrite /mule/= ?mul0r// eqxx. Qed. Canonical mule_mulmonoid := @Monoid.MulLaw _ _ mule mul0e mule0. Definition mule_def x y := ~~ (((x == 0) && (`| y | == +oo)) || ((y == 0) && (`| x | == +oo))). Local Notation "x *? y" := (mule_def x y). Lemma mule_defC x y : x *? y = y *? x. Proof. by rewrite [in LHS]/mule_def orbC. Qed. Lemma mule_def_fin x y : x \is a fin_num -> y \is a fin_num -> x *? y. Proof. move: x y => [x| |] [y| |] finx finy//. by rewrite /mule_def negb_or 2!negb_and/= 2!orbT. Qed. Lemma mule_def_neq0_infty x y : x != 0 -> y \isn't a fin_num -> x *? y. Proof. by move: x y => [x| |] [y| |]// x0 _; rewrite /mule_def (negbTE x0). Qed. Lemma mule_def_infty_neq0 x y : x \isn't a fin_num -> y!= 0 -> x *? y. Proof. by move: x y => [x| |] [y| |]// _ y0; rewrite /mule_def (negbTE y0). Qed. Lemma neq0_mule_def x y : x * y != 0 -> x *? y. Proof. move: x y => [x| |] [y| |] //; first by rewrite mule_def_fin. - by have [->|?] := eqVneq x 0%R; rewrite ?mul0e ?eqxx// mule_def_neq0_infty. - by have [->|?] := eqVneq x 0%R; rewrite ?mul0e ?eqxx// mule_def_neq0_infty. - by have [->|?] := eqVneq y 0%R; rewrite ?mule0 ?eqxx// mule_def_infty_neq0. - by have [->|?] := eqVneq y 0%R; rewrite ?mule0 ?eqxx// mule_def_infty_neq0. Qed. Lemma ltpinfty_adde_def : {in [pred x | x < +oo] &, forall x y, x +? y}. Proof. by move=> [x| |] [y| |]. Qed. Lemma ltninfty_adde_def : {in [pred x | -oo < x] &, forall x y, x +? y}. Proof. by move=> [x| |] [y| |]. Qed. Lemma abse_eq0 x : (`|x| == 0) = (x == 0). Proof. by move: x => [| |] //= r; rewrite !eqe normr_eq0. Qed. Lemma abse0 : `|0| = 0 :> \bar R. Proof. by rewrite /abse normr0. Qed. Lemma abse1 : `|1| = 1 :> \bar R. Proof. by rewrite /abse normr1. Qed. Lemma abseN x : `|- x| = `|x|. Proof. by case: x => [r||]; rewrite //= normrN. Qed. Lemma eqe_opp x y : (- x == - y) = (x == y). Proof. move: x y => [r| |] [r'| |] //=; apply/idP/idP => [|/eqP[->]//]. by move/eqP => -[] /eqP; rewrite eqr_opp => /eqP ->. Qed. Lemma eqe_oppP x y : (- x = - y) <-> (x = y). Proof. by split=> [/eqP | -> //]; rewrite eqe_opp => /eqP. Qed. Lemma eqe_oppLR x y : (- x == y) = (x == - y). Proof. by move: x y => [r0| |] [r1| |] //=; rewrite !eqe eqr_oppLR. Qed. Lemma eqe_oppLRP x y : (- x = y) <-> (x = - y). Proof. split=> /eqP; first by rewrite eqe_oppLR => /eqP. by rewrite -eqe_oppLR => /eqP. Qed. Lemma oppe_subset (A B : set (\bar R)) : ((A `<=` B) <-> (-%E @` A `<=` -%E @` B))%classic. Proof. split=> [AB _ [] x ? <-|AB x Ax]; first by exists x => //; exact: AB. have /AB[y By] : ((-%E @` A) (- x))%classic by exists x. by rewrite eqe_oppP => <-. Qed. Lemma fin_numN x : (- x \is a fin_num) = (x \is a fin_num). Proof. by rewrite !fin_numE 2!eqe_oppLR andbC. Qed. Lemma oppeB x y : y \is a fin_num -> - (x - y) = - x + y. Proof. by move=> yfin; rewrite oppeD ?oppeK// fin_numN. Qed. Lemma fin_numD x y : (x + y \is a fin_num) = (x \is a fin_num) && (y \is a fin_num). Proof. by move: x y => [x| |] [y| |]. Qed. Lemma sum_fin_num (T : Type) (s : seq T) (P : pred T) (f : T -> \bar R) : \sum_(i <- s | P i) f i \is a fin_num = all [pred x | x \is a fin_num] [seq f i | i <- s & P i]. Proof. by rewrite -big_all big_map big_filter; exact: (big_morph _ fin_numD). Qed. Lemma sum_fin_numP (T : eqType) (s : seq T) (P : pred T) (f : T -> \bar R) : reflect (forall i, i \in s -> P i -> f i \is a fin_num) (\sum_(i <- s | P i) f i \is a fin_num). Proof. rewrite sum_fin_num; apply: (iffP allP) => /=. by move=> + x xs Px; apply; rewrite map_f// mem_filter Px. by move=> + _ /mapP[x /[!mem_filter]/andP[Px xs] ->]; apply. Qed. Lemma fin_numB x y : (x - y \is a fin_num) = (x \is a fin_num) && (y \is a fin_num). Proof. by move: x y => [x| |] [y| |]. Qed. Lemma fin_numM x y : x \is a fin_num -> y \is a fin_num -> x * y \is a fin_num. Proof. by move: x y => [x| |] [y| |]. Qed. Lemma fineD : {in @fin_num R &, {morph fine : x y / x + y >-> (x + y)%R}}. Proof. by move=> [r| |] [s| |]. Qed. Lemma fineB : {in @fin_num R &, {morph fine : x y / x - y >-> (x - y)%R}}. Proof. by move=> [r| |] [s| |]. Qed. Lemma fineM : {in @fin_num R &, {morph fine : x y / x * y >-> (x * y)%R}}. Proof. by move=> [x| |] [y| |]. Qed. Lemma fin_num_adde_def x y : y \is a fin_num -> x +? y. Proof. by move: x y => [x| |] [y | |]. Qed. Lemma fineK x : x \is a fin_num -> (fine x)%:E = x. Proof. by case: x. Qed. Lemma telescope_sume n m (f : nat -> \bar R) : (forall i, (n <= i <= m)%N -> f i \is a fin_num) -> (n <= m)%N -> \sum_(n <= k < m) (f k.+1 - f k) = f m - f n. Proof. move=> nmf nm; under eq_big_nat => i /andP[ni im] do rewrite -[f i.+1]fineK -1?[f i]fineK ?(nmf, ni, im) 1?ltnW//= -EFinD. by rewrite sumEFin telescope_sumr// EFinB !fineK ?nmf ?nm ?leqnn. Qed. Lemma addeK x y : x \is a fin_num -> y + x - x = y. Proof. by move: x y => [x| |] [y| |] //; rewrite -EFinD -EFinB addrK. Qed. Lemma subeK x y : y \is a fin_num -> x - y + y = x. Proof. by move: x y => [x| |] [y| |] //; rewrite -EFinD subrK. Qed. Lemma subee x : x \is a fin_num -> x - x = 0. Proof. by move: x => [r _| |] //; rewrite -EFinB subrr. Qed. Lemma sube_eq x y z : x \is a fin_num -> (y +? z) -> (x - z == y) = (x == y + z). Proof. by move: x y z => [x| |] [y| |] [z| |] // _ _; rewrite !eqe subr_eq. Qed. Lemma adde_eq_ninfty x y : (x + y == -oo) = ((x == -oo) || (y == -oo)). Proof. by move: x y => [?| |] [?| |]. Qed. Lemma addye x : x != -oo -> +oo + x = +oo. Proof. by case: x. Qed. Lemma addey x : x != -oo -> x + +oo = +oo. Proof. by case: x. Qed. Lemma adde_Neq_pinfty x y : x != -oo -> y != -oo -> (x + y != +oo) = (x != +oo) && (y != +oo). Proof. by move: x y => [x| |] [y| |]. Qed. Lemma adde_Neq_ninfty x y : x != +oo -> y != +oo -> (x + y != -oo) = (x != -oo) && (y != -oo). Proof. by move: x y => [x| |] [y| |]. Qed. Lemma adde_ss_eq0 x y : (0 <= x) && (0 <= y) || (x <= 0) && (y <= 0) -> x + y == 0 = (x == 0) && (y == 0). Proof. by move=> /orP[|] /andP[]; [exact: padde_eq0|exact: nadde_eq0]. Qed. Lemma esum_ninftyP (T : eqType) (s : seq T) (P : pred T) (f : T -> \bar R) : \sum_(i <- s | P i) f i = -oo <-> exists i, [/\ i \in s, P i & f i = -oo]. Proof. split=> [|[i [si Pi fi]]]. rewrite big_seq_cond; elim/big_ind: _ => // [[?| |] [?| |]//|]. by move=> i /andP[si Pi] fioo; exists i; rewrite si Pi fioo. rewrite big_mkcond (bigID (xpred1 i))/= (eq_bigr (fun _ => -oo)); last first. by move=> j /eqP ->; rewrite Pi. rewrite big_const_seq/= [X in X + _](_ : _ = -oo)//. elim: s i Pi fi si => // h t ih i Pi fi. rewrite inE => /predU1P[<-/=|it]; first by rewrite eqxx. by rewrite /= iterD ih//=; case: (_ == _). Qed. Lemma esum_ninfty (I : finType) (f : I -> \bar R) (P : {pred I}) : (\sum_(i | P i) f i == -oo) = [exists i in P, f i == -oo]. Proof. apply/idP/idP => [/eqP/esum_ninftyP|/existsP[i /andP[Pi /eqP fi]]]. by move=> -[i [_ Pi fi]]; apply/existsP; exists i; rewrite fi eqxx andbT. by apply/eqP/esum_ninftyP; exists i. Qed. Lemma esum_pinftyP (T : eqType) (s : seq T) (P : pred T) (f : T -> \bar R) : (forall i, P i -> f i != -oo) -> \sum_(i <- s | P i) f i = +oo <-> exists i, [/\ i \in s, P i & f i = +oo]. Proof. move=> finoo; split=> [|[i [si Pi fi]]]. rewrite big_seq_cond; elim/big_ind: _ => // [[?| |] [?| |]//|]. by move=> i /andP[si Pi] fioo; exists i; rewrite si Pi fioo. elim: s i Pi fi si => // h t ih i Pi fi. rewrite inE => /predU1P[<-/=|it]. rewrite big_cons Pi fi addye//. by apply/eqP => /esum_ninftyP[j [jt /finoo/negbTE/eqP]]. by rewrite big_cons; case: ifPn => Ph; rewrite (ih i)// addey// finoo. Qed. Lemma esum_pinfty (I : finType) (P : {pred I}) (f : I -> \bar R) : (forall i, P i -> f i != -oo) -> (\sum_(i | P i) f i == +oo) = [exists i in P, f i == +oo]. Proof. move=> fio; apply/idP/existsP => [/eqP /=|[/= i /andP[Pi /eqP fi]]]. have {}fio : (forall i, P i -> f i != -oo) by move=> i Pi; exact: fio. by move=> /(esum_pinftyP _ fio)[i [_ Pi fi]]; exists i; rewrite fi eqxx andbT. by apply/eqP/esum_pinftyP => //; exists i. Qed. Lemma adde_ge0 x y : 0 <= x -> 0 <= y -> 0 <= x + y. Proof. by move: x y => [r0| |] [r1| |] // ? ?; rewrite !lee_fin addr_ge0. Qed. Lemma adde_le0 x y : x <= 0 -> y <= 0 -> x + y <= 0. Proof. move: x y => [r0||] [r1||]// ? ?; rewrite !lee_fin -(addr0 0%R); exact: ler_add. Qed. Lemma oppe_gt0 x : (0 < - x) = (x < 0). Proof. move: x => [x||] //; exact: oppr_gt0. Qed. Lemma oppe_lt0 x : (- x < 0) = (0 < x). Proof. move: x => [x||] //; exact: oppr_lt0. Qed. Lemma oppe_ge0 x : (0 <= - x) = (x <= 0). Proof. move: x => [x||] //; exact: oppr_ge0. Qed. Lemma oppe_le0 x : (- x <= 0) = (0 <= x). Proof. move: x => [x||] //; exact: oppr_le0. Qed. Lemma sume_ge0 T (f : T -> \bar R) (P : pred T) : (forall t, P t -> 0 <= f t) -> forall l, 0 <= \sum_(i <- l | P i) f i. Proof. by move=> f0 l; elim/big_rec : _ => // t x Pt; apply/adde_ge0/f0. Qed. Lemma sume_le0 T (f : T -> \bar R) (P : pred T) : (forall t, P t -> f t <= 0) -> forall l, \sum_(i <- l | P i) f i <= 0. Proof. by move=> f0 l; elim/big_rec : _ => // t x Pt; apply/adde_le0/f0. Qed. Lemma mulNyy : -oo * +oo = -oo :> \bar R. Proof. by rewrite /mule /= lt0y. Qed. Lemma mulyNy : +oo * -oo = -oo :> \bar R. Proof. by rewrite muleC mulNyy. Qed. Lemma mulyy : +oo * +oo = +oo :> \bar R. Proof. by rewrite /mule /= lt0y. Qed. Lemma mulNyNy : -oo * -oo = +oo :> \bar R. Proof. by []. Qed. Lemma real_mulry r : r \is Num.real -> r%:E * +oo = (Num.sg r)%:E * +oo. Proof. move=> rreal; rewrite /mule/= !eqe sgr_eq0; case: ifP => [//|rn0]. move: rreal => /orP[|]; rewrite le_eqVlt. by rewrite eq_sym rn0/= => rgt0; rewrite lte_fin rgt0 gtr0_sg// lte01. by rewrite rn0/= => rlt0; rewrite lt_def lt_geF// andbF ltr0_sg// lte0N1. Qed. Lemma real_mulyr r : r \is Num.real -> +oo * r%:E = (Num.sg r)%:E * +oo. Proof. by move=> rreal; rewrite muleC real_mulry. Qed. Lemma real_mulrNy r : r \is Num.real -> r%:E * -oo = (Num.sg r)%:E * -oo. Proof. move=> rreal; rewrite /mule/= !eqe sgr_eq0; case: ifP => [//|rn0]. move: rreal => /orP[|]; rewrite le_eqVlt. by rewrite eq_sym rn0/= => rgt0; rewrite lte_fin rgt0 gtr0_sg// lte01. by rewrite rn0/= => rlt0; rewrite lt_def lt_geF// andbF ltr0_sg// lte0N1. Qed. Lemma real_mulNyr r : r \is Num.real -> -oo * r%:E = (Num.sg r)%:E * -oo. Proof. by move=> rreal; rewrite muleC real_mulrNy. Qed. Definition real_mulr_infty := (real_mulry, real_mulyr, real_mulrNy, real_mulNyr). Lemma mulN1e x : - 1%E * x = - x. Proof. rewrite -EFinN /mule/=; case: x => [x||]; do ?[by rewrite mulN1r|by rewrite eqe oppr_eq0 oner_eq0 lte_fin ltr0N1]. Qed. Lemma muleN1 x : x * - 1%E = - x. Proof. by rewrite muleC mulN1e. Qed. Lemma mule_neq0 x y : x != 0 -> y != 0 -> x * y != 0. Proof. move: x y => [x||] [y||] x0 y0 //; rewrite /mule/= ?(lt0y,mulf_neq0)//; try by (rewrite (negbTE x0); case: ifPn) || by (rewrite (negbTE y0); case: ifPn). Qed. Lemma mule_eq0 x y : (x * y == 0) = (x == 0) || (y == 0). Proof. apply/idP/idP => [|/orP[] /eqP->]; rewrite ?(mule0, mul0e)//. by apply: contraTT => /norP[]; apply: mule_neq0. Qed. Lemma mule_ge0 x y : 0 <= x -> 0 <= y -> 0 <= x * y. Proof. move: x y => [x||] [y||]//=; rewrite /mule/= ?(lee_fin, eqe, lte_fin, lt0y)//. - exact: mulr_ge0. - rewrite le_eqVlt => /predU1P[<- _|x0 _]; first by rewrite eqxx. by rewrite gt_eqF // x0 le0y. - move=> _; rewrite le_eqVlt => /predU1P[<-|y0]; first by rewrite eqxx. by rewrite gt_eqF // y0 le0y. Qed. Lemma mule_gt0 x y : 0 < x -> 0 < y -> 0 < x * y. Proof. by rewrite !lt_def => /andP[? ?] /andP[? ?]; rewrite mule_neq0 ?mule_ge0. Qed. Lemma mule_le0 x y : x <= 0 -> y <= 0 -> 0 <= x * y. Proof. move: x y => [x||] [y||]//=; rewrite /mule/= ?(lee_fin, eqe, lte_fin)//. - exact: mulr_le0. - by rewrite lt_leAnge => -> _; case: ifP => _ //; rewrite andbF le0y. - by rewrite lt_leAnge => _ ->; case: ifP => _ //; rewrite andbF le0y. Qed. Lemma mule_le0_ge0 x y : x <= 0 -> 0 <= y -> x * y <= 0. Proof. move: x y => [x| |] [y| |] //; rewrite /mule/= ?(lee_fin, lte_fin). - exact: mulr_le0_ge0. - by move=> x0 _; case: ifP => _ //; rewrite lt_leAnge /= x0 andbF leNy0. - move=> _; rewrite le_eqVlt => /predU1P[<-|->]; first by rewrite eqxx. by case: ifP => _ //; rewrite leNy0. - by rewrite lt0y leNy0. Qed. Lemma mule_ge0_le0 x y : 0 <= x -> y <= 0 -> x * y <= 0. Proof. by move=> x0 y0; rewrite muleC mule_le0_ge0. Qed. Lemma mule_lt0_lt0 x y : x < 0 -> y < 0 -> 0 < x * y. Proof. by rewrite !lt_neqAle => /andP[? ?]/andP[*]; rewrite eq_sym mule_neq0 ?mule_le0. Qed. Lemma mule_gt0_lt0 x y : 0 < x -> y < 0 -> x * y < 0. Proof. rewrite lt_def !lt_neqAle => /andP[? ?]/andP[? ?]. by rewrite mule_neq0 ?mule_ge0_le0. Qed. Lemma mule_lt0_gt0 x y : x < 0 -> 0 < y -> x * y < 0. Proof. by move=> x0 y0; rewrite muleC mule_gt0_lt0. Qed. Lemma gte_opp x : 0 < x -> - x < x. Proof. by case: x => //= r; rewrite !lte_fin; apply: gtr_opp. Qed. Lemma realMe x y : (0%E >=< x)%O -> (0%E >=< y)%O -> (0%E >=< x * y)%O. Proof. case: x y => [x||] [y||]// rx ry; do ?[exact: realM |by rewrite /mule/= lt0y |by rewrite real_mulr_infty ?realE -?lee_fin// /Num.sg; case: ifP; [|case: ifP]; rewrite ?mul0e /Order.comparable ?lexx; rewrite ?mulN1e ?leNy0 ?mul1e ?le0y |by rewrite mulNyNy /Order.comparable le0y]. Qed. Lemma abse_ge0 x : 0 <= `|x|. Proof. by move: x => [x| |] /=; rewrite ?le0y ?lee_fin. Qed. Lemma gee0_abs x : 0 <= x -> `|x| = x. Proof. by move: x => [x| |]//; rewrite lee_fin => x0; apply/eqP; rewrite eqe ger0_norm. Qed. Lemma gte0_abs x : 0 < x -> `|x| = x. Proof. by move=> /ltW/gee0_abs. Qed. Lemma lee0_abs x : x <= 0 -> `|x| = - x. Proof. by move: x => [x| |]//; rewrite lee_fin => x0; apply/eqP; rewrite eqe ler0_norm. Qed. Lemma lte0_abs x : x < 0 -> `|x| = - x. Proof. by move=> /ltW/lee0_abs. Qed. End ERealArithTh_numDomainType. Notation "x +? y" := (adde_def x%dE y%dE) : ereal_dual_scope. Notation "x +? y" := (adde_def x y) : ereal_scope. Notation "x *? y" := (mule_def x%dE y%dE) : ereal_dual_scope. Notation "x *? y" := (mule_def x y) : ereal_scope. Notation maxe := (@Order.max ereal_display _). Notation "@ 'maxe' R" := (@Order.max ereal_display R) (at level 10, R at level 8, only parsing) : fun_scope. Notation mine := (@Order.min ereal_display _). Notation "@ 'mine' R" := (@Order.min ereal_display R) (at level 10, R at level 8, only parsing) : fun_scope. Module DualAddTheoryNumDomain. Section DualERealArithTh_numDomainType. Local Open Scope ereal_dual_scope. Context {R : numDomainType}. Implicit Types x y z : \bar R. Lemma dual_addeE x y : (x + y)%dE = - ((- x) + (- y))%E. Proof. by case: x => [x| |]; case: y => [y| |] //=; rewrite opprD !opprK. Qed. Lemma dual_sumeE I (r : seq I) (P : pred I) (F : I -> \bar R) : (\sum_(i <- r | P i) F i)%dE = - (\sum_(i <- r | P i) (- F i)%E)%E. Proof. apply: (big_ind2 (fun x y => x = - y)%E) => [|_ x _ y -> ->|i _]. - by rewrite oppe0. - by rewrite dual_addeE !oppeK. - by rewrite oppeK. Qed. Lemma dual_addeE_def x y : x +? y -> (x + y)%dE = (x + y)%E. Proof. by case: x => [x| |]; case: y. Qed. Lemma dEFinD (r r' : R) : (r + r')%R%:E = r%:E + r'%:E. Proof. by []. Qed. Lemma dEFinB (r r' : R) : (r - r')%R%:E = r%:E - r'%:E. Proof. by []. Qed. Lemma dsumEFin I r P (F : I -> R) : \sum_(i <- r | P i) (F i)%:E = (\sum_(i <- r | P i) F i)%R%:E. Proof. by rewrite dual_sumeE sumEFin sumrN EFinN oppeK. Qed. Lemma daddeC : commutative (S := \bar R) +%dE. Proof. by move=> x y; rewrite !dual_addeE addeC. Qed. Lemma dadde0 : right_id (0 : \bar R) +%dE. Proof. by move=> x; rewrite dual_addeE eqe_oppLRP oppe0 adde0. Qed. Lemma dadd0e : left_id (0 : \bar R) +%dE. Proof. by move=> x;rewrite dual_addeE eqe_oppLRP oppe0 add0e. Qed. Lemma daddeA : associative (S := \bar R) +%dE. Proof. by move=> x y z; rewrite !dual_addeE !oppeK addeA. Qed. Canonical dadde_monoid := Monoid.Law daddeA dadd0e dadde0. Canonical dadde_comoid := Monoid.ComLaw daddeC. Lemma daddeAC : right_commutative (S := \bar R) +%dE. Proof. exact: Monoid.mulmAC. Qed. Lemma daddeCA : left_commutative (S := \bar R) +%dE. Proof. exact: Monoid.mulmCA. Qed. Lemma daddeACA : @interchange (\bar R) +%dE +%dE. Proof. exact: Monoid.mulmACA. Qed. Lemma realDed x y : (0%E >=< x)%O -> (0%E >=< y)%O -> (0%E >=< x + y)%O. Proof. case: x y => [x||] [y||] //; exact: realD. Qed. Lemma doppeD x y : y \is a fin_num -> - (x + y) = - x - y. Proof. by move: y => [y| |] _ //; rewrite !dual_addeE !oppeK oppeD. Qed. Lemma dsube0 x : x - 0 = x. Proof. by move: x => [x| |] //; rewrite -dEFinB subr0. Qed. Lemma dsub0e x : 0 - x = - x. Proof. by move: x => [x| |] //; rewrite -dEFinB sub0r. Qed. Lemma doppeB x y : y \is a fin_num -> - (x - y) = - x + y. Proof. by move=> yfin; rewrite doppeD ?oppeK// fin_numN. Qed. Lemma dfin_numD x y : (x + y \is a fin_num) = (x \is a fin_num) && (y \is a fin_num). Proof. by move: x y => [x| |] [y| |]. Qed. Lemma dfineD : {in (@fin_num R) &, {morph fine : x y / x + y >-> (x + y)%R}}. Proof. by move=> [r| |] [s| |]. Qed. Lemma dfineB : {in @fin_num R &, {morph fine : x y / x - y >-> (x - y)%R}}. Proof. by move=> [r| |] [s| |]. Qed. Lemma daddeK x y : x \is a fin_num -> y + x - x = y. Proof. by move=> fx; rewrite !dual_addeE oppeK addeK ?oppeK; case: x fx. Qed. Lemma dsubeK x y : y \is a fin_num -> x - y + y = x. Proof. by move=> fy; rewrite !dual_addeE oppeK subeK ?oppeK; case: y fy. Qed. Lemma dsubee x : x \is a fin_num -> x - x = 0. Proof. by move=> fx; rewrite dual_addeE subee ?oppe0; case: x fx. Qed. Lemma dsube_eq x y z : x \is a fin_num -> (y +? z) -> (x - z == y) = (x == y + z). Proof. by move: x y z => [x| |] [y| |] [z| |] // _ _; rewrite !eqe subr_eq. Qed. Lemma dadde_eq_pinfty x y : (x + y == +oo) = ((x == +oo) || (y == +oo)). Proof. by move: x y => [?| |] [?| |]. Qed. Lemma daddooe x : x != -oo -> +oo + x = +oo. Proof. by case: x. Qed. Lemma daddeoo x : x != -oo -> x + +oo = +oo. Proof. by case: x. Qed. Lemma dadde_Neq_pinfty x y : x != -oo -> y != -oo -> (x + y != +oo) = (x != +oo) && (y != +oo). Proof. by move: x y => [x| |] [y| |]. Qed. Lemma dadde_Neq_ninfty x y : x != +oo -> y != +oo -> (x + y != -oo) = (x != -oo) && (y != -oo). Proof. by move: x y => [x| |] [y| |]. Qed. Lemma ndadde_eq0 x y : x <= 0 -> y <= 0 -> x + y == 0 = (x == 0) && (y == 0). Proof. move: x y => [x||] [y||] //. - by rewrite !lee_fin -dEFinD !eqe; exact: naddr_eq0. - by rewrite /adde/= (_ : -oo == 0 = false)// andbF. Qed. Lemma pdadde_eq0 x y : 0 <= x -> 0 <= y -> x + y == 0 = (x == 0) && (y == 0). Proof. move: x y => [x||] [y||] //. - by rewrite !lee_fin -dEFinD !eqe; exact: paddr_eq0. - by rewrite /adde/= (_ : +oo == 0 = false)// andbF. Qed. Lemma dadde_ss_eq0 x y : (0 <= x) && (0 <= y) || (x <= 0) && (y <= 0) -> x + y == 0 = (x == 0) && (y == 0). Proof. move=> /orP[|] /andP[]; [exact: pdadde_eq0|exact: ndadde_eq0]. Qed. Lemma desum_pinftyP (T : eqType) (s : seq T) (P : pred T) (f : T -> \bar R) : \sum_(i <- s | P i) f i = +oo <-> exists i, [/\ i \in s, P i & f i = +oo]. Proof. rewrite dual_sumeE eqe_oppLRP /= esum_ninftyP. by split=> -[i + /ltac:(exists i)] => [|] []; [|split]; rewrite // eqe_oppLRP. Qed. Lemma desum_pinfty (I : finType) (f : I -> \bar R) (P : {pred I}) : (\sum_(i | P i) f i == +oo) = [exists i in P, f i == +oo]. Proof. rewrite dual_sumeE eqe_oppLR esum_ninfty. by under eq_existsb => i do rewrite eqe_oppLR. Qed. Lemma desum_ninftyP (T : eqType) (s : seq T) (P : pred T) (f : T -> \bar R) : (forall i, P i -> f i != +oo) -> \sum_(i <- s | P i) f i = -oo <-> exists i, [/\ i \in s, P i & f i = -oo]. Proof. move=> fioo. rewrite dual_sumeE eqe_oppLRP /= esum_pinftyP => [|i Pi]; last first. by rewrite eqe_oppLR fioo. by split=> -[i + /ltac:(exists i)] => [|] []; [|split]; rewrite // eqe_oppLRP. Qed. Lemma desum_ninfty (I : finType) (f : I -> \bar R) (P : {pred I}) : (forall i, f i != +oo) -> (\sum_(i | P i) f i == -oo) = [exists i in P, f i == -oo]. Proof. move=> finoo. rewrite dual_sumeE eqe_oppLR /= esum_pinfty => [|i]; rewrite ?eqe_oppLR //. by under eq_existsb => i do rewrite eqe_oppLR. Qed. Lemma dadde_ge0 x y : 0 <= x -> 0 <= y -> 0 <= x + y. Proof. rewrite dual_addeE oppe_ge0 -!oppe_le0; exact: adde_le0. Qed. Lemma dadde_le0 x y : x <= 0 -> y <= 0 -> x + y <= 0. Proof. rewrite dual_addeE oppe_le0 -!oppe_ge0; exact: adde_ge0. Qed. Lemma dsume_ge0 T (f : T -> \bar R) (P : pred T) : (forall n, P n -> 0 <= f n) -> forall l, 0 <= \sum_(i <- l | P i) f i. Proof. move=> u0 l; rewrite dual_sumeE oppe_ge0 sume_le0 // => t Pt. rewrite oppe_le0; exact: u0. Qed. Lemma dsume_le0 T (f : T -> \bar R) (P : pred T) : (forall n, P n -> f n <= 0) -> forall l, \sum_(i <- l | P i) f i <= 0. Proof. move=> u0 l; rewrite dual_sumeE oppe_le0 sume_ge0 // => t Pt. rewrite oppe_ge0; exact: u0. Qed. Lemma gte_dopp (r : \bar R) : (0 < r)%E -> (- r < r)%E. Proof. by case: r => //= r; rewrite !lte_fin; apply: gtr_opp. Qed. Lemma ednatmul_pinfty n : +oo *+ n.+1 = +oo :> \bar R. Proof. by elim: n => //= n ->. Qed. Lemma ednatmul_ninfty n : -oo *+ n.+1 = -oo :> \bar R. Proof. by elim: n => //= n ->. Qed. Lemma EFin_dnatmul (r : R) n : (r *+ n.+1)%:E = r%:E *+ n.+1. Proof. by elim: n => //= n <-. Qed. Lemma ednatmulE x n : x *+ n = (x *+ n)%E. Proof. case: x => [x| |]; case: n => [//|n]. - by rewrite -EFin_natmul -EFin_dnatmul. - by rewrite enatmul_pinfty ednatmul_pinfty. - by rewrite enatmul_ninfty ednatmul_ninfty. Qed. Lemma dmule2n x : x *+ 2 = x + x. Proof. by []. Qed. End DualERealArithTh_numDomainType. End DualAddTheoryNumDomain. Section ERealArithTh_realDomainType. Context {R : realDomainType}. Implicit Types (x y z u a b : \bar R) (r : R). Lemma fin_numElt x : (x \is a fin_num) = (-oo < x < +oo). Proof. by rewrite fin_numE -leye_eq -leeNy_eq -2!ltNge. Qed. Lemma fin_numPlt x : reflect (-oo < x < +oo) (x \is a fin_num). Proof. by rewrite fin_numElt; exact: idP. Qed. Lemma lte_pinfty_eq x : (x < +oo) = (x \is a fin_num) || (x == -oo). Proof. by case: x => // x //=; exact: ltey. Qed. Lemma ge0_fin_numE x : 0 <= x -> (x \is a fin_num) = (x < +oo). Proof. by move: x => [x| |] => // x0; rewrite fin_numElt ltNye. Qed. Lemma eq_pinftyP x : x = +oo <-> (forall A, (0 < A)%R -> A%:E <= x). Proof. split=> [-> // A A0|Ax]; first by rewrite leey. apply/eqP; rewrite eq_le leey /= leNgt; apply/negP. move: x Ax => [x Ax _|//|]; last by move/(_ _ ltr01). move/not_forallP : Ax; apply; exists (`|x| + 1)%R. apply/not_implyP; split; first by rewrite -(addr0 0%R) ler_lt_add. rewrite lee_fin => /le_trans/(_ (ler_norm x)). by apply/negP; rewrite -ltNge ltr_addl. Qed. Lemma seq_psume_eq0 (I : choiceType) (r : seq I) (P : pred I) (F : I -> \bar R) : (forall i, P i -> 0 <= F i)%E -> (\sum_(i <- r | P i) F i == 0)%E = all (fun i => P i ==> (F i == 0%E)) r. Proof. move=> F0; apply/eqP/allP => PF0; last first. rewrite big_seq_cond big1// => i /andP[ir Pi]. by have := PF0 _ ir; rewrite Pi implyTb => /eqP. move=> i ir; apply/implyP => Pi; apply/eqP. have rPF : {in r, forall i, P i ==> (F i \is a fin_num)}. move=> j jr; apply/implyP => Pj; rewrite fin_numElt; apply/andP; split. by rewrite (lt_le_trans _ (F0 _ Pj))// ltNye. rewrite ltNge; apply/eqP; rewrite leye_eq; apply/eqP/negP => /eqP Fjoo. have PFninfty k : P k -> F k != -oo%E. by move=> Pk; rewrite gt_eqF// (lt_le_trans _ (F0 _ Pk))// ltNye. have /esum_pinftyP : exists i, [/\ i \in r, P i & F i = +oo%E] by exists j. by move=> /(_ PFninfty); rewrite PF0. have ? : (\sum_(i <- r | P i) (fine \o F) i == 0)%R. apply/eqP/EFin_inj; rewrite big_seq_cond -sumEFin. rewrite (eq_bigr (fun i0 => F i0)); last first. move=> j /andP[jr Pj] /=; rewrite fineK//. by have := rPF _ jr; rewrite Pj implyTb. by rewrite -big_seq_cond PF0. have /allP/(_ _ ir) : all (fun i => P i ==> ((fine \o F) i == 0))%R r. by rewrite -psumr_eq0// => j Pj/=; apply/le0R/F0. rewrite Pi implyTb => /= => /eqP Fi0. rewrite -(@fineK _ (F i))//; last by have := rPF _ ir; rewrite Pi implyTb. by rewrite Fi0. Qed. Lemma lte_add_pinfty x y : x < +oo -> y < +oo -> x + y < +oo. Proof. by move: x y => -[r [r'| |]| |] // ? ?; rewrite -EFinD ltey. Qed. Lemma lte_sum_pinfty I (s : seq I) (P : pred I) (f : I -> \bar R) : (forall i, P i -> f i < +oo) -> \sum_(i <- s | P i) f i < +oo. Proof. elim/big_ind : _ => [_|x y xoo yoo foo|i ?]; [exact: ltey| |exact]. by apply: lte_add_pinfty; [exact: xoo| exact: yoo]. Qed. Lemma sube_gt0 x y : (0 < y - x) = (x < y). Proof. by move: x y => [r | |] [r'| |] //=; rewrite ?(ltey, ltNye)// !lte_fin subr_gt0. Qed. Lemma sube_le0 x y : (y - x <= 0) = (y <= x). Proof. by rewrite !leNgt sube_gt0. Qed. Lemma suber_ge0 y x : y \is a fin_num -> (0 <= x - y) = (y <= x). Proof. by move: x y => [x| |] [y| |] //= _; rewrite ?(leey, lee_fin, subr_ge0). Qed. Lemma subre_ge0 x y : y \is a fin_num -> (0 <= y - x) = (x <= y). Proof. by move: x y => [x| |] [y| |] //=; rewrite ?(leey, leNye, lee_fin) //= subr_ge0. Qed. Lemma lte_oppl x y : (- x < y) = (- y < x). Proof. by move: x y => [r| |] [r'| |] //=; rewrite ?(ltey, ltNye)// !lte_fin ltr_oppl. Qed. Lemma lte_oppr x y : (x < - y) = (y < - x). Proof. by move: x y => [r| |] [r'| |] //=; rewrite ?(ltey, ltNye)// !lte_fin ltr_oppr. Qed. Lemma lee_oppr x y : (x <= - y) = (y <= - x). Proof. by move: x y => [r0| |] [r1| |] //=; rewrite ?(leey, leNye)// !lee_fin ler_oppr. Qed. Lemma lee_oppl x y : (- x <= y) = (- y <= x). Proof. by move: x y => [r0| |] [r1| |] //=; rewrite ?(leey, leNye)// !lee_fin ler_oppl. Qed. Lemma muleN x y : x * - y = - (x * y). Proof. move: x y => [x| |] [y| |] //=; rewrite /mule/=; try by rewrite ltey. - by rewrite mulrN. - by rewrite !eqe !lte_fin; case: ltrgtP => //; rewrite oppe0. - by rewrite !eqe !lte_fin; case: ltrgtP => //; rewrite oppe0. - rewrite !eqe oppr_eq0 eq_sym; case: ifP; rewrite ?oppe0// => y0. by rewrite [RHS]fun_if ltNge if_neg EFinN lee_oppl oppe0 le_eqVlt eqe y0. - rewrite !eqe oppr_eq0 eq_sym; case: ifP; rewrite ?oppe0// => y0. by rewrite [RHS]fun_if ltNge if_neg EFinN lee_oppl oppe0 le_eqVlt eqe y0. Qed. Lemma mulNe x y : - x * y = - (x * y). Proof. by rewrite muleC muleN muleC. Qed. Lemma muleNN x y : - x * - y = x * y. Proof. by rewrite mulNe muleN oppeK. Qed. Lemma mulry r : r%:E * +oo%E = (Num.sg r)%:E * +oo%E. Proof. by rewrite [LHS]real_mulry// num_real. Qed. Lemma mulyr r : +oo%E * r%:E = (Num.sg r)%:E * +oo%E. Proof. by rewrite muleC mulry. Qed. Lemma mulrNy r : r%:E * -oo%E = (Num.sg r)%:E * -oo%E. Proof. by rewrite [LHS]real_mulrNy// num_real. Qed. Lemma mulNyr r : -oo%E * r%:E = (Num.sg r)%:E * -oo%E. Proof. by rewrite muleC mulrNy. Qed. Definition mulr_infty := (mulry, mulyr, mulrNy, mulNyr). Lemma lte_mul_pinfty x y : 0 <= x -> x \is a fin_num -> y < +oo -> x * y < +oo. Proof. move: x y => [x| |] [y| |] // x0 xfin _; first by rewrite -EFinM ltey. rewrite mulr_infty; move: x0; rewrite lee_fin le_eqVlt => /predU1P[<-|x0]. - by rewrite sgr0 mul0e ltey. - by rewrite gtr0_sg // mul1e. Qed. Lemma mule_ge0_gt0 x y : 0 <= x -> 0 <= y -> (0 < x * y) = (0 < x) && (0 < y). Proof. move: x y => [x| |] [y| |] //; rewrite ?lee_fin. - by move=> x0 y0; rewrite !lte_fin; exact: mulr_ge0_gt0. - rewrite le_eqVlt => /predU1P[<-|x0] _; first by rewrite mul0e ltxx. by rewrite ltey andbT mulr_infty gtr0_sg// mul1e lte_fin x0 ltey. - move=> _; rewrite le_eqVlt => /predU1P[<-|x0]. by rewrite mule0 ltxx andbC. by rewrite ltey/= mulr_infty gtr0_sg// mul1e lte_fin x0 ltey. - by move=> _ _; rewrite mulyy ltey. Qed. Lemma gt0_mulye x : (0 < x -> +oo * x = +oo)%E. Proof. move: x => [x|_|//]; last by rewrite mulyy. by rewrite lte_fin => x0; rewrite muleC mulr_infty gtr0_sg// mul1e. Qed. Lemma lt0_mulye x : (x < 0 -> +oo * x = -oo)%E. Proof. move: x => [x|//|_]; last by rewrite mulyNy. by rewrite lte_fin => x0; rewrite muleC mulr_infty ltr0_sg// mulN1e. Qed. Lemma gt0_mulNye x : (0 < x -> -oo * x = -oo)%E. Proof. move: x => [x|_|//]; last by rewrite mulNyy. by rewrite lte_fin => x0; rewrite muleC mulr_infty gtr0_sg// mul1e. Qed. Lemma lt0_mulNye x : (x < 0 -> -oo * x = +oo)%E. Proof. move: x => [x|//|_]; last by rewrite mulNyNy. by rewrite lte_fin => x0; rewrite muleC mulr_infty ltr0_sg// mulN1e. Qed. Lemma mule_eq_pinfty x y : (x * y == +oo) = [|| (x > 0) && (y == +oo), (x < 0) && (y == -oo), (x == +oo) && (y > 0) | (x == -oo) && (y < 0)]. Proof. move: x y => [x| |] [y| |]; rewrite ?(lte_fin,andbF,andbT,orbF,eqxx,andbT)//=. - by rewrite mulr_infty; have [/ltr0_sg|/gtr0_sg|] := ltgtP x 0%R; move=> ->; rewrite ?(mulN1e,mul1e,sgr0,mul0e). - by rewrite mulr_infty; have [/ltr0_sg|/gtr0_sg|] := ltgtP x 0%R; move=> ->; rewrite ?(mulN1e,mul1e,sgr0,mul0e). - by rewrite mulr_infty; have [/ltr0_sg|/gtr0_sg|] := ltgtP y 0%R; move=> ->; rewrite ?(mulN1e,mul1e,sgr0,mul0e). - by rewrite mulyy ltey. - by rewrite mulyNy. - by rewrite mulr_infty; have [/ltr0_sg|/gtr0_sg|] := ltgtP y 0%R; move=> ->; rewrite ?(mulN1e,mul1e,sgr0,mul0e). - by rewrite mulNyy. - by rewrite ltNye. Qed. Lemma mule_eq_ninfty x y : (x * y == -oo) = [|| (x > 0) && (y == -oo), (x < 0) && (y == +oo), (x == -oo) && (y > 0) | (x == +oo) && (y < 0)]. Proof. have := mule_eq_pinfty x (- y); rewrite muleN eqe_oppLR => ->. by rewrite !eqe_oppLR lte_oppr lte_oppl oppe0 (orbC _ ((x == -oo) && _)). Qed. Lemma lte_opp x y : (- x < - y) = (y < x). Proof. by rewrite lte_oppl oppeK. Qed. Lemma lte_add a b x y : a < b -> x < y -> a + x < b + y. Proof. move: a b x y=> [a| |] [b| |] [x| |] [y| |]; rewrite ?(ltey,ltNye)//. by rewrite !lte_fin; exact: ltr_add. Qed. Lemma lee_addl x y : 0 <= y -> x <= x + y. Proof. move: x y => -[ x [y| |]//= | [| |]// | [| | ]//]; by [rewrite !lee_fin ler_addl | move=> _; exact: leey]. Qed. Lemma lee_addr x y : 0 <= y -> x <= y + x. Proof. by rewrite addeC; exact: lee_addl. Qed. Lemma gee_addl x y : y <= 0 -> x + y <= x. Proof. move: x y => -[ x [y| |]//= | [| |]// | [| | ]//]; by [rewrite !lee_fin ger_addl | move=> _; exact: leNye]. Qed. Lemma gee_addr x y : y <= 0 -> y + x <= x. Proof. rewrite addeC; exact: gee_addl. Qed. Lemma lte_addl y x : y \is a fin_num -> 0 < x -> y < y + x. Proof. by move: x y => [x| |] [y| |] _ //; rewrite ?ltey ?ltNye // !lte_fin ltr_addl. Qed. Lemma lte_addr y x : y \is a fin_num -> 0 < x -> y < x + y. Proof. rewrite addeC; exact: lte_addl. Qed. Lemma gte_subl y x : y \is a fin_num -> 0 < x -> y - x < y. Proof. move: y x => [x| |] [y| |] _ //; rewrite addeC /= ?ltNye //. by rewrite !lte_fin gtr_addr ltr_oppl oppr0. Qed. Lemma gte_subr y x : y \is a fin_num -> 0 < x -> - x + y < y. Proof. rewrite addeC; exact: gte_subl. Qed. Lemma lte_add2lE x a b : x \is a fin_num -> (x + a < x + b) = (a < b). Proof. move: a b x => [a| |] [b| |] [x| |] _ //; rewrite ?(ltey, ltNye)//. by rewrite !lte_fin ltr_add2l. Qed. Lemma lee_add2l x a b : a <= b -> x + a <= x + b. Proof. move: a b x => -[a [b [x /=|//|//] | []// |//] | []// | ]. - by rewrite !lee_fin ler_add2l. - by move=> r _; exact: leey. - by move=> -[b [| |]// | []// | //] r oob; exact: leNye. Qed. Lemma lee_add2lE x a b : x \is a fin_num -> (x + a <= x + b) = (a <= b). Proof. move: a b x => [a| |] [b| |] [x| |] _ //; rewrite ?(leey, leNye)//. by rewrite !lee_fin ler_add2l. Qed. Lemma lee_add2r x a b : a <= b -> a + x <= b + x. Proof. rewrite addeC (addeC b); exact: lee_add2l. Qed. Lemma lee_add a b x y : a <= b -> x <= y -> a + x <= b + y. Proof. move: a b x y => [a| |] [b| |] [x| |] [y| |]; rewrite ?(leey, leNye)//. by rewrite !lee_fin; exact: ler_add. Qed. Lemma lte_le_add a b x y : b \is a fin_num -> a < x -> b <= y -> a + b < x + y. Proof. move: x y a b => [x| |] [y| |] [a| |] [b| |] _ //=; rewrite ?(ltey, ltNye)//. by rewrite !lte_fin; exact: ltr_le_add. Qed. Lemma lee_lt_add a b x y : a \is a fin_num -> a <= x -> b < y -> a + b < x + y. Proof. by move=> afin xa yb; rewrite (addeC a) (addeC x) lte_le_add. Qed. Lemma lee_sub x y z u : x <= y -> u <= z -> x - z <= y - u. Proof. move: x y z u => -[x| |] -[y| |] -[z| |] -[u| |] //=; rewrite ?(leey,leNye)//. by rewrite !lee_fin; exact: ler_sub. Qed. Lemma lte_le_sub z u x y : u \is a fin_num -> x < z -> u <= y -> x - y < z - u. Proof. move: z u x y => [z| |] [u| |] [x| |] [y| |] _ //=; rewrite ?(ltey, ltNye)//. by rewrite !lte_fin => xltr tley; apply: ltr_le_add; rewrite // ler_oppl opprK. Qed. Lemma lte_pmul2r z : z \is a fin_num -> 0 < z -> {mono *%E^~ z : x y / x < y}. Proof. move: z => [z| |] _ // z0 [x| |] [y| |] //. - by rewrite !lte_fin ltr_pmul2r. - by rewrite mulr_infty gtr0_sg// mul1e 2!ltey. - by rewrite mulr_infty gtr0_sg// mul1e ltNge leNye ltNge leNye. - by rewrite mulr_infty gtr0_sg// mul1e ltNge leey ltNge leey. - by rewrite mulr_infty gtr0_sg// mul1e mulr_infty gtr0_sg// mul1e. - by rewrite mulr_infty gtr0_sg// mul1e 2!ltNye. - by rewrite mulr_infty gtr0_sg// mul1e mulr_infty gtr0_sg// mul1e. Qed. Lemma lte_pmul2l z : z \is a fin_num -> 0 < z -> {mono *%E z : x y / x < y}. Proof. by move=> zfin z0 x y; rewrite 2!(muleC z) lte_pmul2r. Qed. Lemma lte_nmul2l z : z \is a fin_num -> z < 0 -> {mono *%E z : x y /~ x < y}. Proof. rewrite -oppe0 lte_oppr => zfin z0 x y. by rewrite -(oppeK z) !mulNe lte_oppl oppeK -2!mulNe lte_pmul2l ?fin_numN. Qed. Lemma lte_nmul2r z : z \is a fin_num -> z < 0 -> {mono *%E^~ z : x y /~ x < y}. Proof. by move=> zfin z0 x y; rewrite -!(muleC z) lte_nmul2l. Qed. Lemma lee_sum I (f g : I -> \bar R) s (P : pred I) : (forall i, P i -> f i <= g i) -> \sum_(i <- s | P i) f i <= \sum_(i <- s | P i) g i. Proof. by move=> Pfg; elim/big_ind2 : _ => // *; apply lee_add. Qed. Lemma lee_sum_nneg_subset I (s : seq I) (P Q : {pred I}) (f : I -> \bar R) : {subset Q <= P} -> {in [predD P & Q], forall i, 0 <= f i} -> \sum_(i <- s | Q i) f i <= \sum_(i <- s | P i) f i. Proof. move=> QP PQf; rewrite big_mkcond [leRHS]big_mkcond lee_sum// => i. by move/implyP: (QP i); move: (PQf i); rewrite !inE -!topredE/=; do !case: ifP. Qed. Lemma lee_sum_npos_subset I (s : seq I) (P Q : {pred I}) (f : I -> \bar R) : {subset Q <= P} -> {in [predD P & Q], forall i, f i <= 0} -> \sum_(i <- s | P i) f i <= \sum_(i <- s | Q i) f i. Proof. move=> QP PQf; rewrite big_mkcond [leRHS]big_mkcond lee_sum// => i. by move/implyP: (QP i); move: (PQf i); rewrite !inE -!topredE/=; do !case: ifP. Qed. Lemma lee_sum_nneg (I : eqType) (s : seq I) (P Q : pred I) (f : I -> \bar R) : (forall i, P i -> ~~ Q i -> 0 <= f i) -> \sum_(i <- s | P i && Q i) f i <= \sum_(i <- s | P i) f i. Proof. move=> PQf; rewrite [leRHS](bigID Q) /= -[leLHS]adde0 lee_add //. by rewrite sume_ge0// => i /andP[]; exact: PQf. Qed. Lemma lee_sum_npos (I : eqType) (s : seq I) (P Q : pred I) (f : I -> \bar R) : (forall i, P i -> ~~ Q i -> f i <= 0) -> \sum_(i <- s | P i) f i <= \sum_(i <- s | P i && Q i) f i. Proof. move=> PQf; rewrite [leLHS](bigID Q) /= -[leRHS]adde0 lee_add //. by rewrite sume_le0// => i /andP[]; exact: PQf. Qed. Lemma lee_sum_nneg_ord (f : nat -> \bar R) (P : pred nat) : (forall n, P n -> 0 <= f n) -> {homo (fun n => \sum_(i < n | P i) (f i)) : i j / (i <= j)%N >-> i <= j}. Proof. move=> f0 i j le_ij; rewrite (big_ord_widen_cond j) // big_mkcondr /=. by rewrite lee_sum // => k ?; case: ifP => // _; exact: f0. Qed. Lemma lee_sum_npos_ord (f : nat -> \bar R) (P : pred nat) : (forall n, P n -> f n <= 0)%E -> {homo (fun n => \sum_(i < n | P i) (f i)) : i j / (i <= j)%N >-> j <= i}. Proof. move=> f0 m n ?; rewrite [leRHS](big_ord_widen_cond n) // big_mkcondr /=. by rewrite lee_sum // => i ?; case: ifP => // _; exact: f0. Qed. Lemma lee_sum_nneg_natr (f : nat -> \bar R) (P : pred nat) m : (forall n, (m <= n)%N -> P n -> 0 <= f n) -> {homo (fun n => \sum_(m <= i < n | P i) (f i)) : i j / (i <= j)%N >-> i <= j}. Proof. move=> f0 i j le_ij; rewrite -[m]add0n !big_addn !big_mkord. apply: (@lee_sum_nneg_ord (fun k => f (k + m)%N) (fun k => P (k + m)%N)); by [move=> n /f0; apply; rewrite leq_addl | rewrite leq_sub2r]. Qed. Lemma lee_sum_npos_natr (f : nat -> \bar R) (P : pred nat) m : (forall n, (m <= n)%N -> P n -> f n <= 0) -> {homo (fun n => \sum_(m <= i < n | P i) (f i)) : i j / (i <= j)%N >-> j <= i}. Proof. move=> f0 i j le_ij; rewrite -[m]add0n !big_addn !big_mkord. apply: (@lee_sum_npos_ord (fun k => f (k + m)%N) (fun k => P (k + m)%N)); by [move=> n /f0; apply; rewrite leq_addl | rewrite leq_sub2r]. Qed. Lemma lee_sum_nneg_natl (f : nat -> \bar R) (P : pred nat) n : (forall m, (m < n)%N -> P m -> 0 <= f m) -> {homo (fun m => \sum_(m <= i < n | P i) (f i)) : i j / (i <= j)%N >-> j <= i}. Proof. move=> f0 i j le_ij; rewrite !big_geq_mkord/=. rewrite lee_sum_nneg_subset// => [k | k /and3P[_ /f0->//]]. by rewrite ?inE -!topredE/= => /andP[-> /(leq_trans le_ij)->]. Qed. Lemma lee_sum_npos_natl (f : nat -> \bar R) (P : pred nat) n : (forall m, (m < n)%N -> P m -> f m <= 0) -> {homo (fun m => \sum_(m <= i < n | P i) (f i)) : i j / (i <= j)%N >-> i <= j}. Proof. move=> f0 i j le_ij; rewrite !big_geq_mkord/=. rewrite lee_sum_npos_subset// => [k | k /and3P[_ /f0->//]]. by rewrite ?inE -!topredE/= => /andP[-> /(leq_trans le_ij)->]. Qed. Lemma lee_sum_nneg_subfset (T : choiceType) (A B : {fset T}%fset) (P : pred T) (f : T -> \bar R) : {subset A <= B} -> {in [predD B & A], forall t, P t -> 0 <= f t} -> \sum_(t <- A | P t) f t <= \sum_(t <- B | P t) f t. Proof. move=> AB f0; rewrite [leRHS]big_mkcond (big_fsetID _ (mem A) B) /=. rewrite -[leLHS]adde0 lee_add //. rewrite -big_mkcond /= {1}(_ : A = [fset x in B | x \in A]%fset) //. by apply/fsetP=> t; rewrite !inE /= andbC; case: (boolP (_ \in _)) => // /AB. rewrite big_fset /= big_seq_cond sume_ge0 // => t /andP[tB tA]. by case: ifPn => // Pt; rewrite f0 // !inE tA. Qed. Lemma lee_sum_npos_subfset (T : choiceType) (A B : {fset T}%fset) (P : pred T) (f : T -> \bar R) : {subset A <= B} -> {in [predD B & A], forall t, P t -> f t <= 0} -> \sum_(t <- B | P t) f t <= \sum_(t <- A | P t) f t. Proof. move=> AB f0; rewrite big_mkcond (big_fsetID _ (mem A) B) /=. rewrite -[leRHS]adde0 lee_add //. rewrite -big_mkcond /= {3}(_ : A = [fset x in B | x \in A]%fset) //. by apply/fsetP=> t; rewrite !inE /= andbC; case: (boolP (_ \in _)) => // /AB. rewrite big_fset /= big_seq_cond sume_le0 // => t /andP[tB tA]. by case: ifPn => // Pt; rewrite f0 // !inE tA. Qed. Lemma lte_subl_addr x y z : y \is a fin_num -> (x - y < z) = (x < z + y). Proof. move: x y z => [x| |] [y| |] [z| |] _ //=; rewrite ?ltey ?ltNye //. by rewrite !lte_fin ltr_subl_addr. Qed. Lemma lte_subl_addl x y z : y \is a fin_num -> (x - y < z) = (x < y + z). Proof. by move=> ?; rewrite lte_subl_addr// addeC. Qed. Lemma lte_subr_addr x y z : z \is a fin_num -> (x < y - z) = (x + z < y). Proof. move: x y z => [x| |] [y| |] [z| |] _ //=; rewrite ?ltNye ?ltey //. by rewrite !lte_fin ltr_subr_addr. Qed. Lemma lte_subr_addl x y z : z \is a fin_num -> (x < y - z) = (z + x < y). Proof. by move=> ?; rewrite lte_subr_addr// addeC. Qed. Lemma lte_subel_addr x y z : x \is a fin_num -> (x - y < z) = (x < z + y). Proof. move: x y z => [x| |] [y| |] [z| |] _ //=; rewrite ?ltNye ?ltey //. by rewrite !lte_fin ltr_subl_addr. Qed. Lemma lte_subel_addl x y z : x \is a fin_num -> (x - y < z) = (x < y + z). Proof. by move=> ?; rewrite lte_subel_addr// addeC. Qed. Lemma lte_suber_addr x y z : x \is a fin_num -> (x < y - z) = (x + z < y). Proof. move: x y z => [x| |] [y| |] [z| |] _ //=; rewrite ?ltNye ?ltey //. by rewrite !lte_fin ltr_subr_addr. Qed. Lemma lte_suber_addl x y z : x \is a fin_num -> (x < y - z) = (z + x < y). Proof. by move=> ?; rewrite lte_suber_addr// addeC. Qed. Lemma lee_subl_addr x y z : y \is a fin_num -> (x - y <= z) = (x <= z + y). Proof. move: x y z => [x| |] [y| |] [z| |] _ //=; rewrite ?leey ?leNye //. by rewrite !lee_fin ler_subl_addr. Qed. Lemma lee_subl_addl x y z : y \is a fin_num -> (x - y <= z) = (x <= y + z). Proof. by move=> ?; rewrite lee_subl_addr// addeC. Qed. Lemma lee_subr_addr x y z : z \is a fin_num -> (x <= y - z) = (x + z <= y). Proof. move: y x z => [y| |] [x| |] [z| |] _ //=; rewrite ?leNye ?leey //. by rewrite !lee_fin ler_subr_addr. Qed. Lemma lee_subr_addl x y z : z \is a fin_num -> (x <= y - z) = (z + x <= y). Proof. by move=> ?; rewrite lee_subr_addr// addeC. Qed. Lemma lee_subel_addr x y z : z \is a fin_num -> (x - y <= z) = (x <= z + y). Proof. move: x y z => [x| |] [y| |] [z| |] _ //=; rewrite ?leey ?leNye //. by rewrite !lee_fin ler_subl_addr. Qed. Lemma lee_subel_addl x y z : z \is a fin_num -> (x - y <= z) = (x <= y + z). Proof. by move=> ?; rewrite lee_subel_addr// addeC. Qed. Lemma lee_suber_addr x y z : y \is a fin_num -> (x <= y - z) = (x + z <= y). Proof. move: y x z => [y| |] [x| |] [z| |] _ //=; rewrite ?leNye ?leey //. by rewrite !lee_fin ler_subr_addr. Qed. Lemma lee_suber_addl x y z : y \is a fin_num -> (x <= y - z) = (z + x <= y). Proof. by move=> ?; rewrite lee_suber_addr// addeC. Qed. Lemma pmule_rge0 x y : 0 < x -> (x * y >= 0) = (y >= 0). Proof. move=> x_gt0; apply/idP/idP; last exact/mule_ge0/ltW. by apply: contra_le; apply: mule_gt0_lt0. Qed. Lemma pmule_lge0 x y : 0 < x -> (y * x >= 0) = (y >= 0). Proof. by rewrite muleC; apply: pmule_rge0. Qed. Lemma pmule_rlt0 x y : 0 < x -> (x * y < 0) = (y < 0). Proof. by move=> /pmule_rge0; rewrite !ltNge => ->. Qed. Lemma pmule_llt0 x y : 0 < x -> (y * x < 0) = (y < 0). Proof. by rewrite muleC; apply: pmule_rlt0. Qed. Lemma pmule_rle0 x y : 0 < x -> (x * y <= 0) = (y <= 0). Proof. by move=> xgt0; rewrite -oppe_ge0 -muleN pmule_rge0 ?oppe_ge0. Qed. Lemma pmule_lle0 x y : 0 < x -> (y * x <= 0) = (y <= 0). Proof. by rewrite muleC; apply: pmule_rle0. Qed. Lemma pmule_rgt0 x y : 0 < x -> (x * y > 0) = (y > 0). Proof. by move=> xgt0; rewrite -oppe_lt0 -muleN pmule_rlt0 ?oppe_lt0. Qed. Lemma pmule_lgt0 x y : 0 < x -> (y * x > 0) = (y > 0). Proof. by rewrite muleC; apply: pmule_rgt0. Qed. Lemma nmule_rge0 x y : x < 0 -> (x * y >= 0) = (y <= 0). Proof. by rewrite -oppe_gt0 => /pmule_rle0<-; rewrite mulNe oppe_le0. Qed. Lemma nmule_lge0 x y : x < 0 -> (y * x >= 0) = (y <= 0). Proof. by rewrite muleC; apply: nmule_rge0. Qed. Lemma nmule_rle0 x y : x < 0 -> (x * y <= 0) = (y >= 0). Proof. by rewrite -oppe_gt0 => /pmule_rge0<-; rewrite mulNe oppe_ge0. Qed. Lemma nmule_lle0 x y : x < 0 -> (y * x <= 0) = (y >= 0). Proof. by rewrite muleC; apply: nmule_rle0. Qed. Lemma nmule_rgt0 x y : x < 0 -> (x * y > 0) = (y < 0). Proof. by rewrite -oppe_gt0 => /pmule_rlt0<-; rewrite mulNe oppe_lt0. Qed. Lemma nmule_lgt0 x y : x < 0 -> (y * x > 0) = (y < 0). Proof. by rewrite muleC; apply: nmule_rgt0. Qed. Lemma nmule_rlt0 x y : x < 0 -> (x * y < 0) = (y > 0). Proof. by rewrite -oppe_gt0 => /pmule_rgt0<-; rewrite mulNe oppe_gt0. Qed. Lemma nmule_llt0 x y : x < 0 -> (y * x < 0) = (y > 0). Proof. by rewrite muleC; apply: nmule_rlt0. Qed. Lemma mule_lt0 x y : (x * y < 0) = [&& x != 0, y != 0 & (x < 0) (+) (y < 0)]. Proof. have [xlt0|xgt0|->] := ltgtP x 0; last by rewrite mul0e. by rewrite nmule_rlt0//= -leNgt lt_def. by rewrite pmule_rlt0//= !lt_neqAle andbA andbb. Qed. Lemma muleA : associative ( *%E : \bar R -> \bar R -> \bar R ). Proof. move=> [x||] [y||] [z||] //; rewrite /mule/mule_subdef/= ?(ltey, eqe, lte_fin, mulrA)//=. - case: ltrgtP => [y0|y0|<-{y}]; last by rewrite mulr0 eqxx. case: ltrgtP => [x0|x0|<-{x}]; last by rewrite mul0r eqxx. by rewrite gt_eqF mulr_gt0. by rewrite lt_eqF ?nmulr_rlt0// nmulr_rgt0// ltNge (ltW y0). case: ltrgtP => [x0|x0|<-{x}]; last by rewrite mul0r eqxx. by rewrite lt_eqF ?nmulr_llt0// nmulr_lgt0// ltNge (ltW x0). by rewrite gt_eqF ?nmulr_rgt0// y0. - case: ltrgtP => [y0|y0|<-{y}]; last by rewrite mulr0 eqxx. case: ltrgtP => [x0|x0|<-{x}]; last by rewrite mul0r eqxx. by rewrite gt_eqF mulr_gt0. by rewrite lt_eqF ?nmulr_rlt0// nmulr_rgt0// ltNge (ltW y0). case: ltrgtP => [x0|x0|<-{x}]; last by rewrite mul0r eqxx. by rewrite lt_eqF ?nmulr_llt0// nmulr_lgt0// ltNge (ltW x0). by rewrite gt_eqF ?nmulr_rgt0// y0. - case: ltrgtP => [z0|z0|<-{z}]; try by case: ltrgtP => [x0|x0|_] //; rewrite mul0r. by case: ltrgtP; rewrite !mulr0. - by case: ltrgtP => //=; rewrite ?ltey// eqxx. - by case: ltrgtP => /=; rewrite ?ltey// eqxx. - case: ltrgtP => //= [z0|z0|<-{z}]; try by case: ltrgtP => //=; rewrite mul0r. by case: ltrgtP; rewrite !mulr0. - by case: ltrgtP => //=; rewrite eqxx. - by case: ltrgtP => //=; rewrite ?ltey ?eqxx. - case: (ltrgtP 0 y) => [y0/=|y0/=|<-{y}]; last by rewrite mul0r eqxx. case: (ltrgtP 0 z) => [z0/=|z0|<-{z}]; last by rewrite mulr0 eqxx. by rewrite (negbTE (mulf_neq0 _ _)) ?gt_eqF // mulr_gt0// lte_fin z0. by rewrite lt_eqF ?nmulr_llt0// ltNge nmulr_lle0// (ltW y0). case: (ltrgtP 0 z) => [z0/=|z0|<-{z}]; last by rewrite mulr0 eqxx. by rewrite lt_eqF ?nmulr_rlt0// ltNge nmulr_rle0// (ltW z0). by rewrite gt_eqF nmulr_lgt0// y0. - by case: ltrgtP => //=; rewrite ltey. - by case: ltrgtP; rewrite /= ?ltey// eqxx. - by case: ltrgtP; rewrite /= ?ltey// eqxx. - by case: ltrgtP; rewrite /= ?ltey// eqxx. - case: (ltrgtP 0 y) => [y0/=|y0/=|<-{y}]; last by rewrite mul0r eqxx. case: (ltrgtP 0 z) => [z0/=|z0|<-{z}]; last by rewrite mulr0 eqxx. by rewrite (negbTE (mulf_neq0 _ _)) ?gt_eqF // mulr_gt0// lte_fin z0. by rewrite lt_eqF ?nmulr_llt0// ltNge nmulr_lle0// (ltW y0). case: (ltrgtP 0 z) => [z0/=|z0|<-{z}]; last by rewrite mulr0 eqxx. by rewrite lt_eqF ?nmulr_rlt0// ltNge nmulr_rle0// (ltW z0). by rewrite gt_eqF nmulr_lgt0// y0. - by case: ltrgtP; rewrite ?eqxx// ?mul0e//= ltey. - by case: ltrgtP. - by case: ltrgtP => //= [z0|z0]; rewrite ?eqxx// ltey. - by case: ltrgtP; rewrite ?eqxx// ltey. Qed. Local Open Scope ereal_scope. Canonical mule_monoid := Monoid.Law muleA mul1e mule1. Canonical mule_comoid := Monoid.ComLaw muleC. Lemma muleCA : left_commutative ( *%E : \bar R -> \bar R -> \bar R ). Proof. exact: Monoid.mulmCA. Qed. Lemma muleAC : right_commutative ( *%E : \bar R -> \bar R -> \bar R ). Proof. exact: Monoid.mulmAC. Qed. Lemma muleACA : interchange (@mule R) (@mule R). Proof. exact: Monoid.mulmACA. Qed. Lemma muleDr x y z : x \is a fin_num -> y +? z -> x * (y + z) = x * y + x * z. Proof. rewrite /mule/=; move: x y z => [x| |] [y| |] [z| |] //= _; try (by case: ltgtP => // -[] <-; rewrite ?(mul0r,add0r,adde0)) || (by case: ltgtP => //; rewrite adde0). by rewrite mulrDr. Qed. Lemma muleDl x y z : x \is a fin_num -> y +? z -> (y + z) * x = y * x + z * x. Proof. by move=> ? ?; rewrite -!(muleC x) muleDr. Qed. Lemma muleBr x y z : x \is a fin_num -> y +? - z -> x * (y - z) = x * y - x * z. Proof. by move=> ? yz; rewrite muleDr ?muleN. Qed. Lemma muleBl x y z : x \is a fin_num -> y +? - z -> (y - z) * x = y * x - z * x. Proof. by move=> ? yz; rewrite muleDl// mulNe. Qed. Lemma ge0_muleDl x y z : 0 <= y -> 0 <= z -> (y + z) * x = y * x + z * x. Proof. rewrite /mule/=; move: x y z => [r| |] [s| |] [t| |] //= s0 t0. - by rewrite mulrDl. - by case: ltgtP => // -[] <-; rewrite mulr0 adde0. - by case: ltgtP => // -[] <-; rewrite mulr0 adde0. - by case: ltgtP => //; rewrite adde0. - rewrite !eqe paddr_eq0 //; move: s0; rewrite lee_fin. case: (ltgtP s) => //= [s0|->{s}] _; rewrite ?add0e. + rewrite lte_fin -[in LHS](addr0 0%R) ltr_le_add // lte_fin s0. by case: ltgtP t0 => // [t0|[<-{t}]] _; [rewrite gt_eqF|rewrite eqxx]. + by move: t0; rewrite lee_fin; case: (ltgtP t). - by rewrite ltey; case: ltgtP s0. - by rewrite ltey; case: ltgtP t0. - by rewrite ltey. - rewrite !eqe paddr_eq0 //; move: s0; rewrite lee_fin. case: (ltgtP s) => //= [s0|->{s}] _; rewrite ?add0e. + rewrite lte_fin -[in LHS](addr0 0%R) ltr_le_add // lte_fin s0. by case: ltgtP t0 => // [t0|[<-{t}]]. + by move: t0; rewrite lee_fin; case: (ltgtP t). - by rewrite ltey; case: ltgtP s0. - by rewrite ltey; case: ltgtP s0. - by rewrite ltey; case: ltgtP s0. Qed. Lemma ge0_muleDr x y z : 0 <= y -> 0 <= z -> x * (y + z) = x * y + x * z. Proof. by move=> y0 z0; rewrite !(muleC x) ge0_muleDl. Qed. Lemma le0_muleDl x y z : y <= 0 -> z <= 0 -> (y + z) * x = y * x + z * x. Proof. rewrite /mule/=; move: x y z => [r| |] [s| |] [t| |] //= s0 t0. - by rewrite mulrDl. - by case: ltgtP => // -[] <-; rewrite mulr0 adde0. - by case: ltgtP => // -[] <-; rewrite mulr0 adde0. - by case: ltgtP => //; rewrite adde0. - rewrite !eqe naddr_eq0 //; move: s0; rewrite lee_fin. case: (ltgtP s) => //= [s0|->{s}] _; rewrite ?add0e. + rewrite !lte_fin -[in LHS](addr0 0%R) ltNge ler_add // ?ltW //=. by rewrite !ltNge ltW //. + by case: (ltgtP t). - by rewrite ltey; case: ltgtP s0. - by rewrite ltey; case: ltgtP t0. - by rewrite ltey. - rewrite !eqe naddr_eq0 //; move: s0; rewrite lee_fin. case: (ltgtP s) => //= [s0|->{s}] _; rewrite ?add0e. + rewrite !lte_fin -[in LHS](addr0 0%R) ltNge ler_add // ?ltW //=. by rewrite !ltNge ltW // -lee_fin t0; case: eqP. + by case: (ltgtP t). - by rewrite ltNge s0 /=; case: eqP. - by rewrite ltNge t0 /=; case: eqP. Qed. Lemma le0_muleDr x y z : y <= 0 -> z <= 0 -> x * (y + z) = x * y + x * z. Proof. by move=> y0 z0; rewrite !(muleC x) le0_muleDl. Qed. Lemma gee_pmull y x : y \is a fin_num -> 0 < x -> y <= 1 -> y * x <= x. Proof. move: x y => [x| |] [y| |] _ //=. - by rewrite lte_fin => x0 r0; rewrite /mule/= lee_fin ger_pmull. - by move=> _; rewrite /mule/= eqe => r1; rewrite leey. Qed. Lemma lee_wpmul2r x : 0 <= x -> {homo *%E^~ x : y z / y <= z}. Proof. move: x => [x|_|//]. rewrite lee_fin le_eqVlt => /predU1P[<- y z|x0]; first by rewrite 2!mule0. move=> [y| |] [z| |]//; first by rewrite !lee_fin// ler_pmul2r. - by move=> _; rewrite mulr_infty gtr0_sg// mul1e leey. - by move=> _; rewrite mulr_infty gtr0_sg// mul1e leNye. - by move=> _; rewrite 2!mulr_infty gtr0_sg// 2!mul1e. move=> [y| |] [z| |]//. - rewrite lee_fin => yz. have [z0|z0|] := ltgtP 0%R z. + by rewrite [leRHS]mulr_infty gtr0_sg// mul1e leey. + by rewrite mulr_infty ltr0_sg// ?(le_lt_trans yz)// [leRHS]mulr_infty ltr0_sg. + move=> z0; move: yz; rewrite -z0 mul0e le_eqVlt => /predU1P[->|y0]. by rewrite mul0e. by rewrite mulr_infty ltr0_sg// mulN1e leNye. + by move=> _; rewrite mulyy leey. + by move=> _; rewrite mulNyy leNye. + by move=> _; rewrite mulNyy leNye. Qed. Lemma lee_wpmul2l x : 0 <= x -> {homo *%E x : y z / y <= z}. Proof. by move=> x0 y z yz; rewrite !(muleC x) lee_wpmul2r. Qed. Lemma ge0_sume_distrl (I : Type) (s : seq I) x (P : pred I) (F : I -> \bar R) : (forall i, P i -> 0 <= F i) -> (\sum_(i <- s | P i) F i) * x = \sum_(i <- s | P i) (F i * x). Proof. elim: s x P F => [x P F F0|h t ih x P F F0]; first by rewrite 2!big_nil mul0e. rewrite big_cons; case: ifPn => Ph; last by rewrite big_cons (negbTE Ph) ih. by rewrite ge0_muleDl ?sume_ge0// ?F0// ih// big_cons Ph. Qed. Lemma ge0_sume_distrr (I : Type) (s : seq I) x (P : pred I) (F : I -> \bar R) : (forall i, P i -> 0 <= F i) -> x * (\sum_(i <- s | P i) F i) = \sum_(i <- s | P i) (x * F i). Proof. by move=> F0; rewrite muleC ge0_sume_distrl//; under eq_bigr do rewrite muleC. Qed. Lemma le0_sume_distrl (I : Type) (s : seq I) x (P : pred I) (F : I -> \bar R) : (forall i, P i -> F i <= 0) -> (\sum_(i <- s | P i) F i) * x = \sum_(i <- s | P i) (F i * x). Proof. elim: s x P F => [x P F F0|h t ih x P F F0]; first by rewrite 2!big_nil mul0e. rewrite big_cons; case: ifPn => Ph; last by rewrite big_cons (negbTE Ph) ih. by rewrite le0_muleDl ?sume_le0// ?F0// ih// big_cons Ph. Qed. Lemma le0_sume_distrr (I : Type) (s : seq I) x (P : pred I) (F : I -> \bar R) : (forall i, P i -> F i <= 0) -> x * (\sum_(i <- s | P i) F i) = \sum_(i <- s | P i) (x * F i). Proof. by move=> F0; rewrite muleC le0_sume_distrl//; under eq_bigr do rewrite muleC. Qed. Lemma eq_infty x : (forall r, r%:E <= x) -> x = +oo. Proof. case: x => [x /(_ (x + 1)%R)|//|/(_ 0%R)//]. by rewrite EFinD addeC -lee_subr_addr// subee// lee_fin ler10. Qed. Lemma eq_ninfty x : (forall r, x <= r%:E) -> x = -oo. Proof. move=> *; apply: (can_inj oppeK); apply: eq_infty => r. by rewrite lee_oppr -EFinN. Qed. Lemma lee_opp x y : (- x <= - y) = (y <= x). Proof. by rewrite lee_oppl oppeK. Qed. Lemma lee_abs x : x <= `|x|. Proof. by move: x => [x| |]//=; rewrite lee_fin ler_norm. Qed. Lemma abse_id x : `| `|x| | = `|x|. Proof. by move: x => [x| |] //=; rewrite normr_id. Qed. Lemma preimage_abse_ninfty : (@abse R @^-1` [set -oo])%classic = set0. Proof. rewrite predeqE => t; split => //=; apply/eqP. by rewrite gt_eqF// (lt_le_trans _ (abse_ge0 t))// ltNye. Qed. Lemma lte_absl (x y : \bar R) : (`|x| < y)%E = (- y < x < y)%E. Proof. by move: x y => [x| |] [y| |] //=; rewrite ?lte_fin ?ltey ?ltNye// ltr_norml. Qed. Lemma fin_num_abs x : (x \is a fin_num) = (`| x | < +oo)%E. Proof. by rewrite fin_numElt -(lte_absl _ +oo) lt_neqAle leey andbT. Qed. Lemma eqe_absl x y : (`|x| == y) = ((x == y) || (x == - y)) && (0 <= y). Proof. by move: x y => [x| |] [y| |] //=; rewrite? leey// !eqe eqr_norml lee_fin. Qed. Lemma lee_abs_add x y : `|x + y| <= `|x| + `|y|. Proof. by move: x y => [x| |] [y| |] //; rewrite /abse -EFinD lee_fin ler_norm_add. Qed. Lemma lee_abs_sum (I : Type) (s : seq I) (F : I -> \bar R) (P : pred I) : `|\sum_(i <- s | P i) F i| <= \sum_(i <- s | P i) `|F i|. Proof. elim/big_ind2 : _ => //; first by rewrite abse0. by move=> *; exact/(le_trans (lee_abs_add _ _) (lee_add _ _)). Qed. Lemma lee_abs_sub x y : `|x - y| <= `|x| + `|y|. Proof. by move: x y => [x| |] [y| |] //; rewrite /abse -EFinD lee_fin ler_norm_sub. Qed. Lemma abseM : {morph @abse R : x y / x * y}. Proof. have xoo r : `|r%:E * +oo| = `|r|%:E * +oo. have [r0|r0] := leP 0%R r. by rewrite (ger0_norm r0)// gee0_abs// mule_ge0// leey. rewrite (ltr0_norm r0)// lte0_abs// ?EFinN ?mulNe//. by rewrite mule_lt0 /= eqe lt_eqF//= lte_fin r0. move=> [x| |] [y| |] //=; first by rewrite normrM. - by rewrite -abseN -muleNN abseN -EFinN xoo normrN. - by rewrite muleC xoo muleC. - by rewrite mulyy. - by rewrite mulyy mulyNy. - by rewrite -abseN -muleNN abseN -EFinN xoo normrN. - by rewrite mulyy mulNyy. - by rewrite mulyy. Qed. Lemma maxEFin r1 r2 : maxe r1%:E r2%:E = (Num.max r1 r2)%:E. Proof. by have [ab|ba] := leP r1 r2; [apply/max_idPr; rewrite lee_fin|apply/max_idPl; rewrite lee_fin ltW]. Qed. Lemma minEFin r1 r2 : mine r1%:E r2%:E = (Num.min r1 r2)%:E. Proof. by have [ab|ba] := leP r1 r2; [apply/min_idPl; rewrite lee_fin|apply/min_idPr; rewrite lee_fin ltW]. Qed. Lemma adde_maxl : left_distributive (@adde R) maxe. Proof. move=> x y z; have [xy|yx] := leP x y. by apply/esym/max_idPr; rewrite lee_add2r. by apply/esym/max_idPl; rewrite lee_add2r// ltW. Qed. Lemma adde_maxr : right_distributive (@adde R) maxe. Proof. move=> x y z; have [yz|zy] := leP y z. by apply/esym/max_idPr; rewrite lee_add2l. by apply/esym/max_idPl; rewrite lee_add2l// ltW. Qed. Lemma maxye : left_zero (+oo : \bar R) maxe. Proof. by move=> x; have [|//] := leP +oo x; rewrite leye_eq => /eqP. Qed. Lemma maxey : right_zero (+oo : \bar R) maxe. Proof. by move=> x; rewrite maxC maxye. Qed. Lemma maxNye : left_id (-oo : \bar R) maxe. Proof. by move=> x; have [//|] := leP -oo x; rewrite ltNge leNye. Qed. Lemma maxeNy : right_id (-oo : \bar R) maxe. Proof. by move=> x; rewrite maxC maxNye. Qed. Lemma minNye : left_zero (-oo : \bar R) mine. Proof. by move=> x; have [|//] := leP x -oo; rewrite leeNy_eq => /eqP. Qed. Lemma mineNy : right_zero (-oo : \bar R) mine. Proof. by move=> x; rewrite minC minNye. Qed. Lemma minye : left_id (+oo : \bar R) mine. Proof. by move=> x; have [//|] := leP x +oo; rewrite ltNge leey. Qed. Lemma miney : right_id (+oo : \bar R) mine. Proof. by move=> x; rewrite minC minye. Qed. Lemma oppe_max : {morph -%E : x y / maxe x y >-> mine x y : \bar R}. Proof. move=> [x| |] [y| |] //=. - by rewrite maxEFin minEFin -EFinN oppr_max. - by rewrite maxey mineNy. - by rewrite miney. - by rewrite minNye. - by rewrite maxNye minye. Qed. Lemma oppe_min : {morph -%E : x y / mine x y >-> maxe x y : \bar R}. Proof. by move=> x y; rewrite -[RHS]oppeK oppe_max !oppeK. Qed. Lemma maxeMr z x y : z \is a fin_num -> 0 < z -> z * maxe x y = maxe (z * x) (z * y). Proof. move=> + r0; have [xy|yx|->] := ltgtP x y; last by rewrite maxxx. - by move=> zfin; rewrite /maxe lte_pmul2l // xy. - by move=> zfin; rewrite maxC /maxe lte_pmul2l// yx. Qed. Lemma maxeMl z x y : z \is a fin_num -> 0 < z -> maxe x y * z = maxe (x * z) (y * z). Proof. by move=> zfin z0; rewrite muleC maxeMr// !(muleC z). Qed. Lemma mineMr z x y : z \is a fin_num -> 0 < z -> z * mine x y = mine (z * x) (z * y). Proof. by move=> ? ?; rewrite -eqe_oppP -muleN oppe_min maxeMr// !muleN -oppe_min. Qed. Lemma mineMl z x y : z \is a fin_num -> 0 < z -> mine x y * z = mine (x * z) (y * z). Proof. by move=> zfin z0; rewrite muleC mineMr// !(muleC z). Qed. Lemma lee_pemull x y : 0 <= y -> 1 <= x -> y <= x * y. Proof. move: x y => [x| |] [y| |] //; last by rewrite mulyy. - by rewrite -EFinM 3!lee_fin; exact: ler_pemull. - move=> _; rewrite lee_fin => x1. by rewrite mulr_infty gtr0_sg ?mul1e// (lt_le_trans _ x1). - rewrite lee_fin le_eqVlt => /predU1P[<- _|y0 _]; first by rewrite mule0. by rewrite mulr_infty gtr0_sg// mul1e leey. Qed. Lemma lee_nemull x y : y <= 0 -> 1 <= x -> x * y <= y. Proof. move: x y => [x| |] [y| |] //; last by rewrite mulyNy. - by rewrite -EFinM 3!lee_fin; exact: ler_nemull. - move=> _; rewrite lee_fin => x1. by rewrite mulr_infty gtr0_sg ?mul1e// (lt_le_trans _ x1). - rewrite lee_fin le_eqVlt => /predU1P[-> _|y0 _]; first by rewrite mule0. by rewrite mulr_infty ltr0_sg// mulN1e leNye. Qed. Lemma lee_pemulr x y : 0 <= y -> 1 <= x -> y <= y * x. Proof. by move=> y0 x1; rewrite muleC lee_pemull. Qed. Lemma lee_nemulr x y : y <= 0 -> 1 <= x -> y * x <= y. Proof. by move=> y0 x1; rewrite muleC lee_nemull. Qed. Lemma mule_natl x n : n%:R%:E * x = x *+ n. Proof. elim: n => [|n]; first by rewrite mul0e. move: x => [x| |] ih. - by rewrite -EFinM mulr_natl EFin_natmul. - by rewrite mulry gtr0_sg// mul1e enatmul_pinfty. - by rewrite mulrNy gtr0_sg// mul1e enatmul_ninfty. Qed. Lemma lte_pmul x1 y1 x2 y2 : 0 <= x1 -> 0 <= x2 -> x1 < y1 -> x2 < y2 -> x1 * x2 < y1 * y2. Proof. move: x1 y1 x2 y2 => [x1| |] [y1| |] [x2| |] [y2| |] //; rewrite !(lte_fin,lee_fin). - by move=> *; rewrite ltr_pmul. - move=> x10 x20 xy1 xy2. by rewrite mulry gtr0_sg ?mul1e -?EFinM ?ltey// (le_lt_trans _ xy1). - move=> x10 x20 xy1 xy2. by rewrite mulyr gtr0_sg ?mul1e -?EFinM ?ltey// (le_lt_trans _ xy2). - by move=> *; rewrite mulyy -EFinM ltey. Qed. Lemma lee_pmul x1 y1 x2 y2 : 0 <= x1 -> 0 <= x2 -> x1 <= y1 -> x2 <= y2 -> x1 * x2 <= y1 * y2. Proof. move: x1 y1 x2 y2 => [x1| |] [y1| |] [x2| |] [y2| |] //; rewrite !lee_fin. - exact: ler_pmul. - rewrite le_eqVlt => /predU1P[<- x20 y10 _|x10 x20 xy1 _]. by rewrite mul0e mule_ge0// leey. by rewrite mulr_infty gtr0_sg// ?mul1e ?leey// (lt_le_trans x10). - rewrite le_eqVlt => /predU1P[<- _ y10 _|x10 _ xy1 _]. by rewrite mul0e mule_ge0// leey. rewrite mulr_infty gtr0_sg// mul1e mulr_infty gtr0_sg// ?mul1e//. exact: (lt_le_trans x10). - move=> x10; rewrite le_eqVlt => /predU1P[<- _ y20|x20 _ xy2]. by rewrite mule0 mulr_infty mule_ge0// ?leey// lee_fin sgr_ge0. by rewrite mulr_infty gtr0_sg ?mul1e ?leey// (lt_le_trans x20). - by move=> x10 x20 _ _; rewrite mulyy leey. - rewrite le_eqVlt => /predU1P[<- _ _ _|x10 _ _ _]. by rewrite mulyy mul0e leey. by rewrite mulyy mulr_infty gtr0_sg// mul1e. - move=> _; rewrite le_eqVlt => /predU1P[<- _ y20|x20 _ xy2]. by rewrite mule0 mulr_infty mule_ge0// ?leey// lee_fin sgr_ge0. rewrite mulr_infty gtr0_sg// mul1e mulr_infty gtr0_sg ?mul1e//. exact: (lt_le_trans x20). - move=> _; rewrite le_eqVlt => /predU1P[<- _ _|x20 _ _]. by rewrite mule0 mulyy leey. by rewrite mulr_infty gtr0_sg// mul1e// mulyy. Qed. Lemma lee_pmul2l x : x \is a fin_num -> 0 < x -> {mono *%E x : x y / x <= y}. Proof. move: x => [x _|//|//] /[!(@lte_fin R)] x0 [y| |] [z| |]. - by rewrite -2!EFinM 2!lee_fin ler_pmul2l. - by rewrite mulry gtr0_sg// mul1e 2!leey. - by rewrite mulrNy gtr0_sg// mul1e 2!leeNy_eq. - by rewrite mulry gtr0_sg// mul1e 2!leye_eq. - by rewrite mulry gtr0_sg// mul1e. - by rewrite mulry mulrNy gtr0_sg// mul1e mul1e. - by rewrite mulrNy gtr0_sg// mul1e 2!leNye. - by rewrite mulrNy mulry gtr0_sg// 2!mul1e. - by rewrite mulrNy gtr0_sg// mul1e. Qed. Lemma lee_pmul2r x : x \is a fin_num -> 0 < x -> {mono *%E^~ x : x y / x <= y}. Proof. by move=> xfin x0 y z; rewrite -2!(muleC x) lee_pmul2l. Qed. End ERealArithTh_realDomainType. Arguments lee_sum_nneg_ord {R}. Arguments lee_sum_npos_ord {R}. Arguments lee_sum_nneg_natr {R}. Arguments lee_sum_npos_natr {R}. Arguments lee_sum_nneg_natl {R}. Arguments lee_sum_npos_natl {R}. #[global] Hint Extern 0 (is_true (0 <= `| _ |)%E) => solve [apply: abse_ge0] : core. Module DualAddTheoryRealDomain. Section DualERealArithTh_realDomainType. Import DualAddTheoryNumDomain. Local Open Scope ereal_dual_scope. Context {R : realDomainType}. Implicit Types x y z a b : \bar R. Lemma dsube_lt0 x y : (x - y < 0) = (x < y). Proof. by rewrite dual_addeE oppe_lt0 sube_gt0 lte_opp. Qed. Lemma dsube_ge0 x y : (0 <= y - x) = (x <= y). Proof. by rewrite dual_addeE oppe_ge0 sube_le0 lee_opp. Qed. Lemma dsuber_le0 x y : y \is a fin_num -> (x - y <= 0) = (x <= y). Proof. by move=> ?; rewrite dual_addeE oppe_le0 suber_ge0 ?fin_numN// lee_opp. Qed. Lemma dsubre_le0 y x : y \is a fin_num -> (y - x <= 0) = (y <= x). Proof. by move=> ?; rewrite dual_addeE oppe_le0 subre_ge0 ?fin_numN// lee_opp. Qed. Lemma lte_dadd a b x y : a < b -> x < y -> a + x < b + y. Proof. rewrite !dual_addeE lte_opp -lte_opp -(lte_opp y); exact: lte_add. Qed. Lemma lee_daddl x y : 0 <= y -> x <= x + y. Proof. rewrite dual_addeE lee_oppr -oppe_le0; exact: gee_addl. Qed. Lemma lee_daddr x y : 0 <= y -> x <= y + x. Proof. rewrite dual_addeE lee_oppr -oppe_le0; exact: gee_addr. Qed. Lemma gee_daddl x y : y <= 0 -> x + y <= x. Proof. rewrite dual_addeE lee_oppl -oppe_ge0; exact: lee_addl. Qed. Lemma gee_daddr x y : y <= 0 -> y + x <= x. Proof. rewrite dual_addeE lee_oppl -oppe_ge0; exact: lee_addr. Qed. Lemma lte_daddl y x : y \is a fin_num -> 0 < x -> y < y + x. Proof. rewrite -fin_numN dual_addeE lte_oppr => ? ?; exact: gte_subl. Qed. Lemma lte_daddr y x : y \is a fin_num -> 0 < x -> y < x + y. Proof. rewrite -fin_numN dual_addeE lte_oppr addeC; exact: gte_subl. Qed. Lemma gte_dsubl y x : y \is a fin_num -> 0 < x -> y - x < y. Proof. rewrite -fin_numN dual_addeE lte_oppl oppeK; exact: lte_addl. Qed. Lemma gte_dsubr y x : y \is a fin_num -> 0 < x -> - x + y < y. Proof. rewrite -fin_numN dual_addeE lte_oppl oppeK; exact: lte_addr. Qed. Lemma lte_dadd2lE x a b : x \is a fin_num -> (x + a < x + b) = (a < b). Proof. by move=> ?; rewrite !dual_addeE lte_opp lte_add2lE ?fin_numN// lte_opp. Qed. Lemma lee_dadd2l x a b : a <= b -> x + a <= x + b. Proof. rewrite !dual_addeE lee_opp -lee_opp; exact: lee_add2l. Qed. Lemma lee_dadd2lE x a b : x \is a fin_num -> (x + a <= x + b) = (a <= b). Proof. by move=> ?; rewrite !dual_addeE lee_opp lee_add2lE ?fin_numN// lee_opp. Qed. Lemma lee_dadd2r x a b : a <= b -> a + x <= b + x. Proof. rewrite !dual_addeE lee_opp -lee_opp; exact: lee_add2r. Qed. Lemma lee_dadd a b x y : a <= b -> x <= y -> a + x <= b + y. Proof. rewrite !dual_addeE lee_opp -lee_opp -(lee_opp y); exact: lee_add. Qed. Lemma lte_le_dadd a b x y : b \is a fin_num -> a < x -> b <= y -> a + b < x + y. Proof. rewrite !dual_addeE lte_opp -lte_opp; exact: lte_le_sub. Qed. Lemma lee_lt_dadd a b x y : a \is a fin_num -> a <= x -> b < y -> a + b < x + y. Proof. by move=> afin xa yb; rewrite (daddeC a) (daddeC x) lte_le_dadd. Qed. Lemma lee_dsub x y z t : x <= y -> t <= z -> x - z <= y - t. Proof. rewrite !dual_addeE lee_oppl oppeK -lee_opp !oppeK; exact: lee_add. Qed. Lemma lte_le_dsub z u x y : u \is a fin_num -> x < z -> u <= y -> x - y < z - u. Proof. rewrite !dual_addeE lte_opp !oppeK -lte_opp; exact: lte_le_add. Qed. Lemma lee_dsum I (f g : I -> \bar R) s (P : pred I) : (forall i, P i -> f i <= g i) -> \sum_(i <- s | P i) f i <= \sum_(i <- s | P i) g i. Proof. move=> Pfg; rewrite !dual_sumeE lee_opp. apply: lee_sum => i Pi; rewrite lee_opp; exact: Pfg. Qed. Lemma lee_dsum_nneg_subset I (s : seq I) (P Q : {pred I}) (f : I -> \bar R) : {subset Q <= P} -> {in [predD P & Q], forall i, 0 <= f i} -> \sum_(i <- s | Q i) f i <= \sum_(i <- s | P i) f i. Proof. move=> QP PQf; rewrite !dual_sumeE lee_opp. apply: lee_sum_npos_subset => [//|i iPQ]; rewrite oppe_le0; exact: PQf. Qed. Lemma lee_dsum_npos_subset I (s : seq I) (P Q : {pred I}) (f : I -> \bar R) : {subset Q <= P} -> {in [predD P & Q], forall i, f i <= 0} -> \sum_(i <- s | P i) f i <= \sum_(i <- s | Q i) f i. Proof. move=> QP PQf; rewrite !dual_sumeE lee_opp. apply: lee_sum_nneg_subset => [//|i iPQ]; rewrite oppe_ge0; exact: PQf. Qed. Lemma lee_dsum_nneg (I : eqType) (s : seq I) (P Q : pred I) (f : I -> \bar R) : (forall i, P i -> ~~ Q i -> 0 <= f i) -> \sum_(i <- s | P i && Q i) f i <= \sum_(i <- s | P i) f i. Proof. move=> PQf; rewrite !dual_sumeE lee_opp. apply: lee_sum_npos => i Pi nQi; rewrite oppe_le0; exact: PQf. Qed. Lemma lee_dsum_npos (I : eqType) (s : seq I) (P Q : pred I) (f : I -> \bar R) : (forall i, P i -> ~~ Q i -> f i <= 0) -> \sum_(i <- s | P i) f i <= \sum_(i <- s | P i && Q i) f i. Proof. move=> PQf; rewrite !dual_sumeE lee_opp. apply: lee_sum_nneg => i Pi nQi; rewrite oppe_ge0; exact: PQf. Qed. Lemma lee_dsum_nneg_ord (f : nat -> \bar R) (P : pred nat) : (forall n, P n -> 0 <= f n)%E -> {homo (fun n => \sum_(i < n | P i) (f i)) : i j / (i <= j)%N >-> i <= j}. Proof. move=> f0 m n mlen; rewrite !dual_sumeE lee_opp. apply: (lee_sum_npos_ord (fun i => - f i)%E) => [i Pi|//]. rewrite oppe_le0; exact: f0. Qed. Lemma lee_dsum_npos_ord (f : nat -> \bar R) (P : pred nat) : (forall n, P n -> f n <= 0)%E -> {homo (fun n => \sum_(i < n | P i) (f i)) : i j / (i <= j)%N >-> j <= i}. Proof. move=> f0 m n mlen; rewrite !dual_sumeE lee_opp. apply: (lee_sum_nneg_ord (fun i => - f i)%E) => [i Pi|//]. rewrite oppe_ge0; exact: f0. Qed. Lemma lee_dsum_nneg_natr (f : nat -> \bar R) (P : pred nat) m : (forall n, (m <= n)%N -> P n -> 0 <= f n) -> {homo (fun n => \sum_(m <= i < n | P i) (f i)) : i j / (i <= j)%N >-> i <= j}. Proof. move=> f0 i j le_ij; rewrite !dual_sumeE lee_opp. apply: lee_sum_npos_natr => [n ? ?|//]; rewrite oppe_le0; exact: f0. Qed. Lemma lee_dsum_npos_natr (f : nat -> \bar R) (P : pred nat) m : (forall n, (m <= n)%N -> P n -> f n <= 0) -> {homo (fun n => \sum_(m <= i < n | P i) (f i)) : i j / (i <= j)%N >-> j <= i}. Proof. move=> f0 i j le_ij; rewrite !dual_sumeE lee_opp. apply: lee_sum_nneg_natr => [n ? ?|//]; rewrite oppe_ge0; exact: f0. Qed. Lemma lee_dsum_nneg_natl (f : nat -> \bar R) (P : pred nat) n : (forall m, (m < n)%N -> P m -> 0 <= f m) -> {homo (fun m => \sum_(m <= i < n | P i) (f i)) : i j / (i <= j)%N >-> j <= i}. Proof. move=> f0 i j le_ij; rewrite !dual_sumeE lee_opp. apply: lee_sum_npos_natl => [m ? ?|//]; rewrite oppe_le0; exact: f0. Qed. Lemma lee_dsum_npos_natl (f : nat -> \bar R) (P : pred nat) n : (forall m, (m < n)%N -> P m -> f m <= 0) -> {homo (fun m => \sum_(m <= i < n | P i) (f i)) : i j / (i <= j)%N >-> i <= j}. Proof. move=> f0 i j le_ij; rewrite !dual_sumeE lee_opp. apply: lee_sum_nneg_natl => [m ? ?|//]; rewrite oppe_ge0; exact: f0. Qed. Lemma lee_dsum_nneg_subfset (T : choiceType) (A B : {fset T}%fset) (P : pred T) (f : T -> \bar R) : {subset A <= B} -> {in [predD B & A], forall t, P t -> 0 <= f t} -> \sum_(t <- A | P t) f t <= \sum_(t <- B | P t) f t. Proof. move=> AB f0; rewrite !dual_sumeE lee_opp. apply: lee_sum_npos_subfset => [//|? ? ?]; rewrite oppe_le0; exact: f0. Qed. Lemma lee_dsum_npos_subfset (T : choiceType) (A B : {fset T}%fset) (P : pred T) (f : T -> \bar R) : {subset A <= B} -> {in [predD B & A], forall t, P t -> f t <= 0} -> \sum_(t <- B | P t) f t <= \sum_(t <- A | P t) f t. Proof. move=> AB f0; rewrite !dual_sumeE lee_opp. apply: lee_sum_nneg_subfset => [//|? ? ?]; rewrite oppe_ge0; exact: f0. Qed. Lemma lte_dsubl_addr x y z : y \is a fin_num -> (x - y < z) = (x < z + y). Proof. by move=> ?; rewrite !dual_addeE lte_oppl lte_oppr oppeK lte_subl_addr. Qed. Lemma lte_dsubl_addl x y z : y \is a fin_num -> (x - y < z) = (x < y + z). Proof. by move=> ?; rewrite !dual_addeE lte_oppl lte_oppr lte_subr_addl ?fin_numN. Qed. Lemma lte_dsubr_addr x y z : z \is a fin_num -> (x < y - z) = (x + z < y). Proof. by move=> ?; rewrite !dual_addeE lte_oppl lte_oppr lte_subl_addr ?fin_numN. Qed. Lemma lte_dsubr_addl x y z : z \is a fin_num -> (x < y - z) = (z + x < y). Proof. by move=> ?; rewrite !dual_addeE lte_oppl lte_oppr lte_subl_addl ?fin_numN. Qed. Lemma lte_dsuber_addr x y z : y \is a fin_num -> (x < y - z) = (x + z < y). Proof. by move=> ?; rewrite !dual_addeE lte_oppl lte_oppr lte_subel_addr ?fin_numN. Qed. Lemma lte_dsuber_addl x y z : y \is a fin_num -> (x < y - z) = (z + x < y). Proof. by move=> ?; rewrite !dual_addeE lte_oppl lte_oppr lte_subel_addl ?fin_numN. Qed. Lemma lte_dsubel_addr x y z : z \is a fin_num -> (x - y < z) = (x < z + y). Proof. by move=> ?; rewrite !dual_addeE lte_oppl lte_oppr lte_suber_addr ?fin_numN. Qed. Lemma lte_dsubel_addl x y z : z \is a fin_num -> (x - y < z) = (x < y + z). Proof. by move=> ?; rewrite !dual_addeE lte_oppl lte_oppr lte_suber_addl ?fin_numN. Qed. Lemma lee_dsubl_addr x y z : y \is a fin_num -> (x - y <= z) = (x <= z + y). Proof. by move=> ?; rewrite !dual_addeE lee_oppl lee_oppr lee_subr_addr ?fin_numN. Qed. Lemma lee_dsubl_addl x y z : y \is a fin_num -> (x - y <= z) = (x <= y + z). Proof. by move=> ?; rewrite !dual_addeE lee_oppl lee_oppr lee_subr_addl ?fin_numN. Qed. Lemma lee_dsubr_addr x y z : z \is a fin_num -> (x <= y - z) = (x + z <= y). Proof. by move=> ?; rewrite !dual_addeE lee_oppl lee_oppr lee_subl_addr ?fin_numN. Qed. Lemma lee_dsubr_addl x y z : z \is a fin_num -> (x <= y - z) = (z + x <= y). Proof. by move=> ?; rewrite !dual_addeE lee_oppl lee_oppr lee_subl_addl ?fin_numN. Qed. Lemma lee_dsubel_addr x y z : x \is a fin_num -> (x - y <= z) = (x <= z + y). Proof. by move=> ?; rewrite !dual_addeE lee_oppl lee_oppr lee_suber_addr ?fin_numN. Qed. Lemma lee_dsubel_addl x y z : x \is a fin_num -> (x - y <= z) = (x <= y + z). Proof. by move=> ?; rewrite !dual_addeE lee_oppl lee_oppr lee_suber_addl ?fin_numN. Qed. Lemma lee_dsuber_addr x y z : x \is a fin_num -> (x <= y - z) = (x + z <= y). Proof. by move=> ?; rewrite !dual_addeE lee_oppl lee_oppr lee_subel_addr ?fin_numN. Qed. Lemma lee_dsuber_addl x y z : x \is a fin_num -> (x <= y - z) = (z + x <= y). Proof. by move=> ?; rewrite !dual_addeE lee_oppl lee_oppr lee_subel_addl ?fin_numN. Qed. Lemma dmuleDr x y z : x \is a fin_num -> y +? z -> x * (y + z) = x * y + x * z. Proof. by move=> *; rewrite !dual_addeE muleN muleDr ?adde_defNN// !muleN. Qed. Lemma dmuleDl x y z : x \is a fin_num -> y +? z -> (y + z) * x = y * x + z * x. Proof. by move=> *; rewrite -!(muleC x) dmuleDr. Qed. Lemma dge0_muleDl x y z : 0 <= y -> 0 <= z -> (y + z) * x = y * x + z * x. Proof. by move=> *; rewrite !dual_addeE mulNe le0_muleDl ?oppe_le0 ?mulNe. Qed. Lemma dge0_muleDr x y z : 0 <= y -> 0 <= z -> x * (y + z) = x * y + x * z. Proof. by move=> *; rewrite !dual_addeE muleN le0_muleDr ?oppe_le0 ?muleN. Qed. Lemma dle0_muleDl x y z : y <= 0 -> z <= 0 -> (y + z) * x = y * x + z * x. Proof. by move=> *; rewrite !dual_addeE mulNe ge0_muleDl ?oppe_ge0 ?mulNe. Qed. Lemma dle0_muleDr x y z : y <= 0 -> z <= 0 -> x * (y + z) = x * y + x * z. Proof. by move=> *; rewrite !dual_addeE muleN ge0_muleDr ?oppe_ge0 ?muleN. Qed. Lemma ge0_dsume_distrl (I : Type) (s : seq I) x (P : pred I) (F : I -> \bar R) : (forall i, P i -> 0 <= F i) -> (\sum_(i <- s | P i) F i) * x = \sum_(i <- s | P i) (F i * x). Proof. move=> F0; rewrite !dual_sumeE !mulNe le0_sume_distrl => [|i Pi]. - by under eq_bigr => i _ do rewrite mulNe. - by rewrite oppe_le0 F0. Qed. Lemma ge0_dsume_distrr (I : Type) (s : seq I) x (P : pred I) (F : I -> \bar R) : (forall i, P i -> 0 <= F i) -> x * (\sum_(i <- s | P i) F i) = \sum_(i <- s | P i) (x * F i). Proof. by move=> F0; rewrite muleC ge0_dsume_distrl//; under eq_bigr do rewrite muleC. Qed. Lemma le0_dsume_distrl (I : Type) (s : seq I) x (P : pred I) (F : I -> \bar R) : (forall i, P i -> F i <= 0) -> (\sum_(i <- s | P i) F i) * x = \sum_(i <- s | P i) (F i * x). Proof. move=> F0; rewrite !dual_sumeE mulNe ge0_sume_distrl => [|i Pi]. - by under eq_bigr => i _ do rewrite mulNe. - by rewrite oppe_ge0 F0. Qed. Lemma le0_dsume_distrr (I : Type) (s : seq I) x (P : pred I) (F : I -> \bar R) : (forall i, P i -> F i <= 0) -> x * (\sum_(i <- s | P i) F i) = \sum_(i <- s | P i) (x * F i). Proof. by move=> F0; rewrite muleC le0_dsume_distrl//; under eq_bigr do rewrite muleC. Qed. Lemma lee_abs_dadd x y : `|x + y| <= `|x| + `|y|. Proof. by move: x y => [x| |] [y| |] //; rewrite /abse -dEFinD lee_fin ler_norm_add. Qed. Lemma lee_abs_dsum (I : Type) (s : seq I) (F : I -> \bar R) (P : pred I) : `|\sum_(i <- s | P i) F i| <= \sum_(i <- s | P i) `|F i|. Proof. elim/big_ind2 : _ => //; first by rewrite abse0. by move=> *; exact/(le_trans (lee_abs_dadd _ _) (lee_dadd _ _)). Qed. Lemma lee_abs_dsub x y : `|x - y| <= `|x| + `|y|. Proof. by move: x y => [x| |] [y| |] //; rewrite /abse -dEFinD lee_fin ler_norm_sub. Qed. Lemma dadde_minl : left_distributive (@dual_adde R) mine. Proof. by move=> x y z; rewrite !dual_addeE oppe_min adde_maxl oppe_max. Qed. Lemma dadde_minr : right_distributive (@dual_adde R) mine. Proof. by move=> x y z; rewrite !dual_addeE oppe_min adde_maxr oppe_max. Qed. Lemma dmule_natl x n : n%:R%:E * x = x *+ n. Proof. by rewrite mule_natl ednatmulE. Qed. End DualERealArithTh_realDomainType. Arguments lee_dsum_nneg_ord {R}. Arguments lee_dsum_npos_ord {R}. Arguments lee_dsum_nneg_natr {R}. Arguments lee_dsum_npos_natr {R}. Arguments lee_dsum_nneg_natl {R}. Arguments lee_dsum_npos_natl {R}. End DualAddTheoryRealDomain. Lemma lee_opp2 {R : realDomainType} : {mono @oppe R : x y /~ x <= y}. Proof. move=> x y; case: x y => [?||] [?||] //; first by rewrite !lee_fin !ler_opp2. by rewrite leNye /Order.le /= realN num_real. by rewrite leey /Order.le /= realN num_real. Qed. Lemma lte_opp2 {R : realDomainType} : {mono @oppe R : x y /~ x < y}. Proof. move=> x y; case: x y => [?||] [?||] //; first by rewrite !lte_fin !ltr_opp2. by rewrite ltNye /Order.lt /= realN num_real. by rewrite ltey /Order.lt /= realN num_real. Qed. Section realFieldType_lemmas. Variable R : realFieldType. Implicit Types x y : \bar R. Implicit Types r : R. Lemma lee_adde x y : (forall e : {posnum R}, x <= y + e%:num%:E) -> x <= y. Proof. move: x y => [x| |] [y| |] //=; rewrite ?(leNye, leey) //; try by move/(_ (PosNum ltr01)). rewrite lee_fin => abe; rewrite leNgt; apply/negP => ba; apply/existsNP : abe. have xy : (0 < (x - y) / 2)%R by apply divr_gt0 => //; rewrite subr_gt0. exists (PosNum xy); apply/negP; rewrite -ltNge lte_fin -ltr_subr_addl. by rewrite ltr_pdivr_mulr // ltr_pmulr ?subr_gt0 // ltr1n. Qed. Lemma lee_paddl y x z : 0 <= x -> y <= z -> y <= x + z. Proof. by move=> *; rewrite -[y]add0e lee_add. Qed. Lemma lte_paddl y x z : 0 <= x -> y < z -> y < x + z. Proof. by move=> x0 /lt_le_trans; apply; rewrite lee_paddl. Qed. Lemma lee_paddr y x z : 0 <= x -> y <= z -> y <= z + x. Proof. by move=> *; rewrite addeC lee_paddl. Qed. Lemma lte_paddr y x z : 0 <= x -> y < z -> y < z + x. Proof. by move=> *; rewrite addeC lte_paddl. Qed. Lemma lte_spaddr z x y : z \is a fin_num -> 0 < y -> z <= x -> z < x + y. Proof. move: z y x => [z| |] [y| |] [x| |] _ //=; rewrite ?(lte_fin, lte_fin, ltey) //. exact: ltr_spaddr. Qed. Lemma lee_mul01Pr x y : 0 <= x -> reflect (forall r, (0 < r < 1)%R -> r%:E * x <= y) (x <= y). Proof. move=> x0; apply/(iffP idP) => [xy r /andP[r0 r1]|h]. move: x0 xy; rewrite le_eqVlt => /predU1P[<-|x0 xy]; first by rewrite mule0. by rewrite (le_trans _ xy) // gee_pmull // ltW. have h01 : (0 < (2^-1 : R) < 1)%R by rewrite invr_gt0 ?invf_lt1 ?ltr0n ?ltr1n. move: x y => [x||] [y||] // in x0 h *. - move: (x0); rewrite lee_fin le_eqVlt => /predU1P[<-|{}x0]. by rewrite (le_trans _ (h _ h01))// mule_ge0// lee_fin. have y0 : (0 < y)%R. by rewrite -lte_fin (lt_le_trans _ (h _ h01))// mule_gt0// lte_fin. rewrite lee_fin leNgt; apply/negP => yx. have /h : (0 < (y + x) / (2 * x) < 1)%R. apply/andP; split; first by rewrite divr_gt0 // ?addr_gt0// ?mulr_gt0. by rewrite ltr_pdivr_mulr ?mulr_gt0// mul1r mulr_natl mulr2n ltr_add2r. rewrite -(EFinM _ x) lee_fin invrM ?unitfE// ?gt_eqF// -mulrA mulrAC. by rewrite mulVr ?unitfE ?gt_eqF// mul1r; apply/negP; rewrite -ltNge midf_lt. - by rewrite leey. - by have := h _ h01. - by have := h _ h01; rewrite mulr_infty sgrV gtr0_sg // mul1e. - by have := h _ h01; rewrite mulr_infty sgrV gtr0_sg // mul1e. Qed. Lemma lte_pdivr_mull r x y : (0 < r)%R -> (r^-1%:E * y < x) = (y < r%:E * x). Proof. move=> r0; move: x y => [x| |] [y| |] //=. - by rewrite 2!lte_fin ltr_pdivr_mull. - by rewrite mulr_infty sgrV gtr0_sg// mul1e 2!ltNge 2!leey. - by rewrite mulr_infty sgrV gtr0_sg// mul1e -EFinM 2!ltNye. - by rewrite mulr_infty gtr0_sg// mul1e 2!ltey. - by rewrite mulr_infty [in RHS]mulr_infty sgrV gtr0_sg// mul1e ltxx. - by rewrite mulr_infty [in RHS]mulr_infty sgrV gtr0_sg// 2!mul1e. - by rewrite mulr_infty gtr0_sg// mul1e. - by rewrite mulr_infty [in RHS]mulr_infty sgrV gtr0_sg// 2!mul1e. - by rewrite mulr_infty [in RHS]mulr_infty sgrV gtr0_sg// mul1e. Qed. Lemma lte_pdivr_mulr r x y : (0 < r)%R -> (y * r^-1%:E < x) = (y < x * r%:E). Proof. by move=> r0; rewrite muleC lte_pdivr_mull// muleC. Qed. Lemma lte_pdivl_mull r y x : (0 < r)%R -> (x < r^-1%:E * y) = (r%:E * x < y). Proof. move=> r0; move: x y => [x| |] [y| |] //=. - by rewrite 2!lte_fin ltr_pdivl_mull. - by rewrite mulr_infty sgrV gtr0_sg// mul1e 2!ltey. - by rewrite mulr_infty sgrV gtr0_sg// mul1e. - by rewrite mulr_infty gtr0_sg// mul1e. - by rewrite mulr_infty [in RHS]mulr_infty sgrV gtr0_sg// mul1e. - by rewrite mulr_infty [in RHS]mulr_infty sgrV gtr0_sg// 2!mul1e. - by rewrite mulr_infty gtr0_sg// mul1e 2!ltNye. - by rewrite mulr_infty [in RHS]mulr_infty sgrV gtr0_sg// 2!mul1e. - by rewrite mulr_infty [in RHS]mulr_infty sgrV gtr0_sg// mul1e. Qed. Lemma lte_pdivl_mulr r x y : (0 < r)%R -> (x < y * r^-1%:E) = (x * r%:E < y). Proof. by move=> r0; rewrite muleC lte_pdivl_mull// muleC. Qed. Lemma lte_ndivl_mulr r x y : (r < 0)%R -> (x < y * r^-1%:E) = (y < x * r%:E). Proof. rewrite -oppr0 ltr_oppr => r0; rewrite -{1}(opprK r) invrN. by rewrite EFinN muleN lte_oppr lte_pdivr_mulr// EFinN muleNN. Qed. Lemma lte_ndivl_mull r x y : (r < 0)%R -> (x < r^-1%:E * y) = (y < r%:E * x). Proof. by move=> r0; rewrite muleC lte_ndivl_mulr// muleC. Qed. Lemma lte_ndivr_mull r x y : (r < 0)%R -> (r^-1%:E * y < x) = (r%:E * x < y). Proof. rewrite -oppr0 ltr_oppr => r0; rewrite -{1}(opprK r) invrN. by rewrite EFinN mulNe lte_oppl lte_pdivl_mull// EFinN muleNN. Qed. Lemma lte_ndivr_mulr r x y : (r < 0)%R -> (y * r^-1%:E < x) = (x * r%:E < y). Proof. by move=> r0; rewrite muleC lte_ndivr_mull// muleC. Qed. Lemma lee_pdivr_mull r x y : (0 < r)%R -> (r^-1%:E * y <= x) = (y <= r%:E * x). Proof. move=> r0; apply/idP/idP. - rewrite le_eqVlt => /predU1P[<-|]; last by rewrite lte_pdivr_mull// => /ltW. by rewrite muleA -EFinM divrr ?mul1e// unitfE gt_eqF. - rewrite le_eqVlt => /predU1P[->|]; last by rewrite -lte_pdivr_mull// => /ltW. by rewrite muleA -EFinM mulVr ?mul1e// unitfE gt_eqF. Qed. Lemma lee_pdivr_mulr r x y : (0 < r)%R -> (y * r^-1%:E <= x) = (y <= x * r%:E). Proof. by move=> r0; rewrite muleC lee_pdivr_mull// muleC. Qed. Lemma lee_pdivl_mull r y x : (0 < r)%R -> (x <= r^-1%:E * y) = (r%:E * x <= y). Proof. move=> r0; apply/idP/idP. - rewrite le_eqVlt => /predU1P[->|]; last by rewrite lte_pdivl_mull// => /ltW. by rewrite muleA -EFinM divrr ?mul1e// unitfE gt_eqF. - rewrite le_eqVlt => /predU1P[<-|]; last by rewrite -lte_pdivl_mull// => /ltW. by rewrite muleA -EFinM mulVr ?mul1e// unitfE gt_eqF. Qed. Lemma lee_pdivl_mulr r x y : (0 < r)%R -> (x <= y * r^-1%:E) = (x * r%:E <= y). Proof. by move=> r0; rewrite muleC lee_pdivl_mull// muleC. Qed. Lemma lee_ndivl_mulr r x y : (r < 0)%R -> (x <= y * r^-1%:E) = (y <= x * r%:E). Proof. rewrite -oppr0 ltr_oppr => r0; rewrite -{1}(opprK r) invrN. by rewrite EFinN muleN lee_oppr lee_pdivr_mulr// EFinN muleNN. Qed. Lemma lee_ndivl_mull r x y : (r < 0)%R -> (x <= r^-1%:E * y) = (y <= r%:E * x). Proof. by move=> r0; rewrite muleC lee_ndivl_mulr// muleC. Qed. Lemma lee_ndivr_mull r x y : (r < 0)%R -> (r^-1%:E * y <= x) = (r%:E * x <= y). Proof. rewrite -oppr0 ltr_oppr => r0; rewrite -{1}(opprK r) invrN. by rewrite EFinN mulNe lee_oppl lee_pdivl_mull// EFinN muleNN. Qed. Lemma lee_ndivr_mulr r x y : (r < 0)%R -> (y * r^-1%:E <= x) = (x * r%:E <= y). Proof. by move=> r0; rewrite muleC lee_ndivr_mull// muleC. Qed. End realFieldType_lemmas. Module DualAddTheoryRealField. Import DualAddTheoryNumDomain DualAddTheoryRealDomain. Section DualRealFieldType_lemmas. Local Open Scope ereal_dual_scope. Variable R : realFieldType. Implicit Types x y : \bar R. Lemma lee_dadde x y : (forall e : {posnum R}, x <= y + e%:num%:E) -> x <= y. Proof. by move=> xye; apply: lee_adde => e; case: x {xye} (xye e). Qed. Lemma lee_pdaddl y x z : 0 <= x -> y <= z -> y <= x + z. Proof. by move=> *; rewrite -[y]dadd0e lee_dadd. Qed. Lemma lte_pdaddl y x z : 0 <= x -> y < z -> y < x + z. Proof. by move=> x0 /lt_le_trans; apply; rewrite lee_pdaddl. Qed. Lemma lee_pdaddr y x z : 0 <= x -> y <= z -> y <= z + x. Proof. by move=> *; rewrite daddeC lee_pdaddl. Qed. Lemma lte_pdaddr y x z : 0 <= x -> y < z -> y < z + x. Proof. by move=> *; rewrite daddeC lte_pdaddl. Qed. Lemma lte_spdaddr (r : R) x y : 0 < y -> r%:E <= x -> r%:E < x + y. Proof. move: y x => [y| |] [x| |] //=; rewrite ?lte_fin ?ltt_fin ?ltey //. exact: ltr_spaddr. Qed. End DualRealFieldType_lemmas. End DualAddTheoryRealField. Module DualAddTheory. Export DualAddTheoryNumDomain. Export DualAddTheoryRealDomain. Export DualAddTheoryRealField. End DualAddTheory. Section ereal_supremum. Variable R : realFieldType. Local Open Scope classical_set_scope. Implicit Types (S : set (\bar R)) (x y : \bar R). Lemma ereal_ub_pinfty S : ubound S +oo. Proof. by apply/ubP=> x _; rewrite leey. Qed. Lemma ereal_ub_ninfty S : ubound S -oo -> S = set0 \/ S = [set -oo]. Proof. have [->|/set0P[x Sx] Snoo] := eqVneq S set0; first by left. right; rewrite predeqE => y; split => [/Snoo|->{y}]. by rewrite leeNy_eq => /eqP ->. by have := Snoo _ Sx; rewrite leeNy_eq => /eqP <-. Qed. Lemma ereal_supremums_set0_ninfty : supremums (@set0 (\bar R)) -oo. Proof. by split; [exact/ubP | apply/lbP=> y _; rewrite leNye]. Qed. Lemma supremum_pinfty S x0 : S +oo -> supremum x0 S = +oo. Proof. move=> Spoo; rewrite /supremum ifF; last by apply/eqP => S0; rewrite S0 in Spoo. have sSoo : supremums S +oo. split; first exact: ereal_ub_pinfty. by move=> /= y /(_ _ Spoo); rewrite leye_eq => /eqP ->. case: xgetP. by move=> _ -> sSxget; move: (is_subset1_supremums sSoo sSxget). by move/(_ +oo) => gSoo; exfalso; apply gSoo => {gSoo}. Qed. Definition ereal_sup S := supremum -oo S. Definition ereal_inf S := - ereal_sup (-%E @` S). Lemma ereal_sup0 : ereal_sup set0 = -oo. Proof. exact: supremum0. Qed. Lemma ereal_sup1 x : ereal_sup [set x] = x. Proof. exact: supremum1. Qed. Lemma ereal_inf0 : ereal_inf set0 = +oo. Proof. by rewrite /ereal_inf image_set0 ereal_sup0. Qed. Lemma ereal_inf1 x : ereal_inf [set x] = x. Proof. by rewrite /ereal_inf image_set1 ereal_sup1 oppeK. Qed. Lemma ub_ereal_sup S M : ubound S M -> ereal_sup S <= M. Proof. rewrite /ereal_sup /supremum; case: ifPn => [/eqP ->|]; first by rewrite leNye. - by move=> _ SM; case: xgetP => [_ -> [_]| _] /=; [exact |rewrite leNye]. Qed. Lemma lb_ereal_inf S M : lbound S M -> M <= ereal_inf S. Proof. move=> SM; rewrite /ereal_inf lee_oppr; apply ub_ereal_sup => x [y Sy <-{x}]. by rewrite lee_oppl oppeK; apply SM. Qed. Lemma ub_ereal_sup_adherent S (e : R) : (0 < e)%R -> ereal_sup S \is a fin_num -> exists2 x, S x & (ereal_sup S - e%:E < x). Proof. move=> e0 Sr; have : ~ ubound S (ereal_sup S - e%:E). move/ub_ereal_sup; apply/negP. by rewrite -ltNge lte_subl_addr // lte_addl // lte_fin. move/asboolP; rewrite asbool_neg; case/existsp_asboolPn => /= x. by rewrite not_implyE => -[? ?]; exists x => //; rewrite ltNge; apply/negP. Qed. Lemma lb_ereal_inf_adherent S (e : R) : (0 < e)%R -> ereal_inf S \is a fin_num -> exists2 x, S x & (x < ereal_inf S + e%:E). Proof. move=> e0; rewrite fin_numN => /(ub_ereal_sup_adherent e0)[x []]. move=> y Sy <-; rewrite -lte_oppr => /lt_le_trans ex; exists y => //. by apply: ex; rewrite oppeD// oppeK. Qed. Lemma ereal_sup_gt S x : x < ereal_sup S -> exists2 y, S y & x < y. Proof. rewrite not_exists2P => + g; apply/negP; rewrite -leNgt. by apply: ub_ereal_sup => y Sy; move: (g y) => [//|/negP]; rewrite leNgt. Qed. Lemma ereal_inf_lt S x : ereal_inf S < x -> exists2 y, S y & y < x. Proof. rewrite lte_oppl => /ereal_sup_gt[_ [y Sy <-]]. by rewrite lte_oppl oppeK => xlty; exists y. Qed. End ereal_supremum. Section ereal_supremum_realType. Variable R : realType. Local Open Scope classical_set_scope. Implicit Types S : set (\bar R). Implicit Types x : \bar R. Let fine_def r0 x : R := if x is r%:E then r else r0. (* NB: see also fine above *) Lemma ereal_supremums_neq0 S : supremums S !=set0. Proof. have [->|Snoo] := eqVneq S [set -oo]. by exists -oo; split; [rewrite ub_set1 |exact: lb_ub_refl]. have [->|S0] := eqVneq S set0. by exists -oo; exact: ereal_supremums_set0_ninfty. have [Spoo|Spoo] := pselect (S +oo). by exists +oo; split; [apply/ereal_ub_pinfty | apply/lbP => /= y /ubP; apply]. have [r Sr] : exists r, S r%:E. move: S0 => /set0P[] [r Sr| // |Snoo1]; first by exists r. apply/not_existsP => nS; move/negP : Snoo; apply. by apply/eqP; rewrite predeqE => -[] // r; split => // /nS. set U := fine_def r @` S. have [|] := eqVneq (ubound U) set0. rewrite -subset0 => U0; exists +oo. split; [exact/ereal_ub_pinfty | apply/lbP => /= -[r0 /ubP Sr0|//|]]. - suff : ubound U r0 by move/U0. by apply/ubP=> _ -[] [r1 Sr1 <-|//| /= _ <-]; rewrite -lee_fin; apply Sr0. - by move/ereal_ub_ninfty => [|]; by [move/eqP : S0|move/eqP : Snoo]. set u : R := sup U. exists u%:E; split; last first. apply/lbP=> -[r0 /ubP Sr0| |]. - rewrite lee_fin; apply/sup_le_ub; first by exists r, r%:E. by apply/ubP => _ -[[r2 ? <-| // | /= _ <-]]; rewrite -lee_fin; exact: Sr0. - by rewrite leey. - by move/ereal_ub_ninfty=> [|/eqP //]; [move/eqP : S0|rewrite (negbTE Snoo)]. apply/ubP => -[r0 Sr0|//|_]; last by rewrite leNye. rewrite lee_fin. suff : has_sup U by move/sup_upper_bound/ubP; apply; exists r0%:E. split; first by exists r0, r0%:E. exists u; apply/ubP => y; move=> [] y' Sy' <-{y}. have : has_sup U by split; [exists r, r%:E | exact/set0P]. move/sup_upper_bound/ubP; apply. by case: y' Sy' => [r1 /= Sr1 | // | /= _]; [exists r1%:E | exists r%:E]. Qed. Lemma ereal_sup_ub S : ubound S (ereal_sup S). Proof. move=> y Sy; rewrite /ereal_sup /supremum ifF; last first. by apply/eqP; rewrite predeqE => /(_ y)[+ _]; exact. case: xgetP => /=; first by move=> _ -> -[] /ubP geS _; apply geS. by case: (ereal_supremums_neq0 S) => /= x0 Sx0; move/(_ x0). Qed. Lemma ereal_sup_ninfty S : ereal_sup S = -oo <-> S `<=` [set -oo]. Proof. split. by move=> supS [r /ereal_sup_ub | /ereal_sup_ub |//]; rewrite supS. move=> /(@subset_set1 _ S) [] ->; [exact: ereal_sup0|exact: ereal_sup1]. Qed. Lemma ereal_inf_lb S : lbound S (ereal_inf S). Proof. by move=> x Sx; rewrite /ereal_inf lee_oppl; apply ereal_sup_ub; exists x. Qed. Lemma ereal_inf_pinfty S : ereal_inf S = +oo <-> S `<=` [set +oo]. Proof. rewrite eqe_oppLRP oppe_subset image_set1; exact: ereal_sup_ninfty. Qed. Lemma le_ereal_sup : {homo @ereal_sup R : A B / A `<=` B >-> A <= B}. Proof. by move=> A B AB; apply ub_ereal_sup => x Ax; apply/ereal_sup_ub/AB. Qed. Lemma le_ereal_inf : {homo @ereal_inf R : A B / A `<=` B >-> B <= A}. Proof. by move=> A B AB; apply lb_ereal_inf => x Bx; exact/ereal_inf_lb/AB. Qed. Lemma hasNub_ereal_sup (A : set (\bar R)) : ~ has_ubound A -> A !=set0 -> ereal_sup A = +oo%E. Proof. move=> hasNubA A0. apply/eqP; rewrite eq_le leey /= leNgt; apply: contra_notN hasNubA => Aoo. by exists (ereal_sup A); exact: ereal_sup_ub. Qed. Lemma ereal_sup_EFin (A : set R) : has_ubound A -> A !=set0 -> ereal_sup (EFin @` A) = (sup A)%:E. Proof. move=> has_ubA A0; apply/eqP; rewrite eq_le; apply/andP; split. by apply: ub_ereal_sup => /= y [r Ar <-{y}]; rewrite lee_fin sup_ub. set esup := ereal_sup _; have := leey esup. rewrite le_eqVlt => /predU1P[->|esupoo]; first by rewrite leey. have := leNye esup; rewrite le_eqVlt => /predU1P[/esym|ooesup]. case: A0 => i Ai. by move=> /ereal_sup_ninfty /(_ i%:E) /(_ (ex_intro2 A _ i Ai erefl)). have esup_fin_num : esup \is a fin_num. by rewrite fin_numE -leeNy_eq -ltNge ooesup /= -leye_eq -ltNge esupoo. rewrite -(@fineK _ esup) // lee_fin leNgt. apply/negP => /(sup_gt A0)[r Ar]; apply/negP; rewrite -leNgt. by rewrite -lee_fin fineK//; apply: ereal_sup_ub; exists r. Qed. Lemma ereal_inf_EFin (A : set R) : has_lbound A -> A !=set0 -> ereal_inf (EFin @` A) = (inf A)%:E. Proof. move=> has_lbA A0; rewrite /ereal_inf /inf EFinN; congr (- _)%E. rewrite -ereal_sup_EFin; [|exact/has_lb_ubN|exact/nonemptyN]. by rewrite !image_comp. Qed. End ereal_supremum_realType. Canonical ereal_pointed (R : numDomainType) := PointedType (extended R) 0%E. Lemma restrict_abse T (R : numDomainType) (f : T -> \bar R) (D : set T) : (abse \o f) \_ D = abse \o (f \_ D). Proof. by apply/funext=> t; rewrite /restrict/=; case: ifPn => // _; rewrite abse0. Qed. Definition posnume (R : numDomainType) of phant R := {> 0 : \bar R}. Notation "{ 'posnum' '\bar' R }" := (@posnume _ (Phant R)) : type_scope. Definition nonnege (R : numDomainType) of phant R := {>= 0 : \bar R}. Notation "{ 'nonneg' '\bar' R }" := (@nonnege _ (Phant R)) : type_scope. Notation "x %:pos" := (widen_signed x%:sgn : {posnum \bar _}) (only parsing) : ereal_dual_scope. Notation "x %:pos" := (widen_signed x%:sgn : {posnum \bar _}) (only parsing) : ereal_scope. Notation "x %:nng" := (widen_signed x%:sgn : {nonneg \bar _}) (only parsing) : ereal_dual_scope. Notation "x %:nng" := (widen_signed x%:sgn : {nonneg \bar _}) (only parsing) : ereal_scope. Notation "x %:pos" := (@widen_signed ereal_display _ _ _ _ (@Signed.from _ _ _ _ _ _ (Phantom _ x)) !=0 (KnownSign.Real (KnownSign.Sign >=0)) _ _) (only printing) : ereal_dual_scope. Notation "x %:pos" := (@widen_signed ereal_display _ _ _ _ (@Signed.from _ _ _ _ _ _ (Phantom _ x)) !=0 (KnownSign.Real (KnownSign.Sign >=0)) _ _) (only printing) : ereal_scope. Notation "x %:nng" := (@widen_signed ereal_display _ _ _ _ (@Signed.from _ _ _ _ _ _ (Phantom _ x)) ?=0 (KnownSign.Real (KnownSign.Sign >=0)) _ _) (only printing) : ereal_dual_scope. Notation "x %:nng" := (@widen_signed ereal_display _ _ _ _ (@Signed.from _ _ _ _ _ _ (Phantom _ x)) ?=0 (KnownSign.Real (KnownSign.Sign >=0)) _ _) (only printing) : ereal_scope. #[global] Hint Extern 0 (is_true (0%E < _)%O) => solve [apply: gt0] : core. #[global] Hint Extern 0 (is_true (_ < 0%E)%O) => solve [apply: lt0] : core. #[global] Hint Extern 0 (is_true (0%E <= _)%O) => solve [apply: ge0] : core. #[global] Hint Extern 0 (is_true (_ <= 0%E)%O) => solve [apply: le0] : core. #[global] Hint Extern 0 (is_true (0%E >=< _)%O) => solve [apply: cmp0] : core. #[global] Hint Extern 0 (is_true (_ != 0%E)%O) => solve [apply: neq0] : core. Section SignedNumDomainStability. Context {R : numDomainType}. Lemma pinfty_snum_subproof : Signed.spec 0 !=0 >=0 (+oo : \bar R). Proof. by rewrite /= le0y. Qed. Canonical pinfty_snum := Signed.mk (pinfty_snum_subproof). Lemma ninfty_snum_subproof : Signed.spec 0 !=0 <=0 (-oo : \bar R). Proof. by rewrite /= leNy0. Qed. Canonical ninfty_snum := Signed.mk (ninfty_snum_subproof). Lemma EFin_snum_subproof nz cond (x : {num R & nz & cond}) : Signed.spec 0 nz cond x%:num%:E. Proof. apply/andP; split. case: cond nz x => [[[]|]|] [] x //=; do ?[by case: (bottom x)|by rewrite eqe eq0F]. case: cond nz x => [[[]|]|] [] x //=; do ?[by case: (bottom x)|by rewrite ?lee_fin ?(eq0, ge0, le0) ?[_ || _]cmp0]. Qed. Canonical EFin_snum nz cond (x : {num R & nz & cond}) := Signed.mk (EFin_snum_subproof x). Lemma fine_snum_subproof (xnz : KnownSign.nullity) (xr : KnownSign.reality) (x : {compare (0 : \bar R) & xnz & xr}) : Signed.spec 0%R ?=0 xr (fine x%:num). Proof. case: xr x => [[[]|]|]//= [x /andP[_]]/=. - by move=> /eqP ->. - by case: x. - by case: x. - by move=> /orP[]; case: x => [x||]//=; rewrite lee_fin => ->; rewrite ?orbT. Qed. Canonical fine_snum (xnz : KnownSign.nullity) (xr : KnownSign.reality) (x : {compare (0 : \bar R) & xnz & xr}) := Signed.mk (fine_snum_subproof x). Lemma oppe_snum_subproof (xnz : KnownSign.nullity) (xr : KnownSign.reality) (x : {compare (0 : \bar R) & xnz & xr}) (r := opp_reality_subdef xnz xr) : Signed.spec 0 xnz r (- x%:num). Proof. rewrite {}/r; case: xr xnz x => [[[]|]|] [] x //=; do ?[by case: (bottom x) |by rewrite ?eqe_oppLR ?oppe0 1?eq0//; rewrite ?oppe_le0 ?oppe_ge0 ?(eq0, eq0F, ge0, le0)//; rewrite orbC [_ || _]cmp0]. Qed. Canonical oppe_snum (xnz : KnownSign.nullity) (xr : KnownSign.reality) (x : {compare (0 : \bar R) & xnz & xr}) := Signed.mk (oppe_snum_subproof x). Lemma adde_snum_subproof (xnz ynz : KnownSign.nullity) (xr yr : KnownSign.reality) (x : {compare (0 : \bar R) & xnz & xr}) (y : {compare (0 : \bar R) & ynz & yr}) (rnz := add_nonzero_subdef xnz ynz xr yr) (rrl := add_reality_subdef xnz ynz xr yr) : Signed.spec 0 rnz rrl (x%:num + y%:num). Proof. rewrite {}/rnz {}/rrl; apply/andP; split. move: xr yr xnz ynz x y => [[[]|]|] [[[]|]|] [] []//= x y; by rewrite 1?adde_ss_eq0 ?(eq0F, ge0, le0, andbF, orbT). move: xr yr xnz ynz x y => [[[]|]|] [[[]|]|] [] []//= x y; do ?[by case: (bottom x)|by case: (bottom y) |by rewrite adde_ge0|by rewrite adde_le0 |exact: realDe|by rewrite 2!eq0 add0e]. Qed. Canonical adde_snum (xnz ynz : KnownSign.nullity) (xr yr : KnownSign.reality) (x : {compare (0 : \bar R) & xnz & xr}) (y : {compare (0 : \bar R) & ynz & yr}) := Signed.mk (adde_snum_subproof x y). Import DualAddTheory. Lemma dadde_snum_subproof (xnz ynz : KnownSign.nullity) (xr yr : KnownSign.reality) (x : {compare (0 : \bar R) & xnz & xr}) (y : {compare (0 : \bar R) & ynz & yr}) (rnz := add_nonzero_subdef xnz ynz xr yr) (rrl := add_reality_subdef xnz ynz xr yr) : Signed.spec 0 rnz rrl (x%:num + y%:num)%dE. Proof. rewrite {}/rnz {}/rrl; apply/andP; split. move: xr yr xnz ynz x y => [[[]|]|] [[[]|]|] [] []//= x y; by rewrite 1?dadde_ss_eq0 ?(eq0F, ge0, le0, andbF, orbT). move: xr yr xnz ynz x y => [[[]|]|] [[[]|]|] [] []//= x y; do ?[by case: (bottom x)|by case: (bottom y) |by rewrite dadde_ge0|by rewrite dadde_le0 |exact: realDed|by rewrite 2!eq0 dadd0e]. Qed. Canonical dadde_snum (xnz ynz : KnownSign.nullity) (xr yr : KnownSign.reality) (x : {compare (0 : \bar R) & xnz & xr}) (y : {compare (0 : \bar R) & ynz & yr}) := Signed.mk (dadde_snum_subproof x y). Lemma mule_snum_subproof (xnz ynz : KnownSign.nullity) (xr yr : KnownSign.reality) (x : {compare (0 : \bar R) & xnz & xr}) (y : {compare (0 : \bar R) & ynz & yr}) (rnz := mul_nonzero_subdef xnz ynz xr yr) (rrl := mul_reality_subdef xnz ynz xr yr) : Signed.spec 0 rnz rrl (x%:num * y%:num). Proof. rewrite {}/rnz {}/rrl; apply/andP; split. by move: xr yr xnz ynz x y => [[[]|]|] [[[]|]|] [] []// x y; rewrite mule_neq0. by move: xr yr xnz ynz x y => [[[]|]|] [[[]|]|] [] []/= x y //; do ?[by case: (bottom x)|by case: (bottom y) |by rewrite mule_ge0|by rewrite mule_le0_ge0 |by rewrite mule_ge0_le0|by rewrite mule_le0|exact: realMe |by rewrite eq0 ?mule0// mul0e]. Qed. Canonical mule_snum (xnz ynz : KnownSign.nullity) (xr yr : KnownSign.reality) (x : {compare (0 : \bar R) & xnz & xr}) (y : {compare (0 : \bar R) & ynz & yr}) := Signed.mk (mule_snum_subproof x y). Definition abse_reality_subdef (xnz : KnownSign.nullity) (xr : KnownSign.reality) := (if xr is =0 then =0 else >=0)%snum_sign. Arguments abse_reality_subdef /. Lemma abse_snum_subproof (xnz : KnownSign.nullity) (xr : KnownSign.reality) (x : {compare (0 : \bar R) & xnz & xr}) (r := abse_reality_subdef xnz xr) : Signed.spec 0 xnz r `|x%:num|. Proof. rewrite {}/r; case: xr xnz x => [[[]|]|] [] x //=; do ?[by case: (bottom x)|by rewrite eq0 abse0 |by rewrite abse_ge0// andbT gee0_abs |by rewrite abse_ge0// andbT lee0_abs |by rewrite abse_ge0// andbT abse_eq0]. Qed. Canonical abse_snum (xnz : KnownSign.nullity) (xr : KnownSign.reality) (x : {compare (0 : \bar R) & xnz & xr}) := Signed.mk (abse_snum_subproof x). End SignedNumDomainStability. Section SignedRealFieldStability. Context {R : realFieldType}. Definition ereal_sup_reality_subdef (xnz : KnownSign.nullity) (xr : KnownSign.reality) := (if KnownSign.wider_reality <=0 xr then KnownSign.Real <=0 else >=<0)%snum_sign. Arguments ereal_sup_reality_subdef /. Lemma ereal_sup_snum_subproof (xnz : KnownSign.nullity) (xr : KnownSign.reality) (S : {compare (0 : \bar R) & xnz & xr} -> Prop) (r := ereal_sup_reality_subdef xnz xr) : Signed.spec 0 ?=0 r (ereal_sup [set x%:num | x in S]%classic). Proof. rewrite {}/r; move: xr S => [[[]|]|] S /=; do ?[by apply: ub_ereal_sup => _ [? _ <-] |by case: ereal_sup => [s||]; rewrite ?leey ?leNye// !lee_fin -realE num_real]. Qed. Canonical ereal_sup_snum (xnz : KnownSign.nullity) (xr : KnownSign.reality) (S : {compare (0 : \bar R) & xnz & xr} -> Prop) := Signed.mk (ereal_sup_snum_subproof S). Definition ereal_inf_reality_subdef (xnz : KnownSign.nullity) (xr : KnownSign.reality) := (if KnownSign.wider_reality >=0 xr then KnownSign.Real >=0 else >=<0)%snum_sign. Arguments ereal_inf_reality_subdef /. Lemma ereal_inf_snum_subproof (xnz : KnownSign.nullity) (xr : KnownSign.reality) (S : {compare (0 : \bar R) & xnz & xr} -> Prop) (r := ereal_inf_reality_subdef xnz xr) : Signed.spec 0 ?=0 r (ereal_inf [set x%:num | x in S]%classic). Proof. rewrite {}/r; move: xr S => [[[]|]|] S /=; do ?[by apply: lb_ereal_inf => _ [? _ <-] |by case: ereal_inf => [s||]; rewrite ?leey ?leNye// !lee_fin -realE num_real]. Qed. Canonical ereal_inf_snum (xnz : KnownSign.nullity) (xr : KnownSign.reality) (S : {compare (0 : \bar R) & xnz & xr} -> Prop) := Signed.mk (ereal_inf_snum_subproof S). End SignedRealFieldStability. Section MorphNum. Context {R : numDomainType} {nz : KnownSign.nullity} {cond : KnownSign.reality}. Local Notation nR := {compare (0 : \bar R) & nz & cond}. Implicit Types (a : \bar R). Lemma num_abse_eq0 a : (`|a|%:nng == 0%:nng) = (a == 0). Proof. by rewrite -abse_eq0. Qed. End MorphNum. Section MorphReal. Context {R : numDomainType} {nz : KnownSign.nullity} {r : KnownSign.real}. Local Notation nR := {compare (0 : \bar R) & nz & r}. Implicit Type x y : nR. Local Notation num := (@num _ _ (0 : R) nz r). Lemma num_lee_maxr a x y : a <= maxe x%:num y%:num = (a <= x%:num) || (a <= y%:num). Proof. by rewrite -comparable_le_maxr// ereal_comparable. Qed. Lemma num_lee_maxl a x y : maxe x%:num y%:num <= a = (x%:num <= a) && (y%:num <= a). Proof. by rewrite -comparable_le_maxl// ereal_comparable. Qed. Lemma num_lee_minr a x y : a <= mine x%:num y%:num = (a <= x%:num) && (a <= y%:num). Proof. by rewrite -comparable_le_minr// ereal_comparable. Qed. Lemma num_lee_minl a x y : mine x%:num y%:num <= a = (x%:num <= a) || (y%:num <= a). Proof. by rewrite -comparable_le_minl// ereal_comparable. Qed. Lemma num_lte_maxr a x y : a < maxe x%:num y%:num = (a < x%:num) || (a < y%:num). Proof. by rewrite -comparable_lt_maxr// ereal_comparable. Qed. Lemma num_lte_maxl a x y : maxe x%:num y%:num < a = (x%:num < a) && (y%:num < a). Proof. by rewrite -comparable_lt_maxl// ereal_comparable. Qed. Lemma num_lte_minr a x y : a < mine x%:num y%:num = (a < x%:num) && (a < y%:num). Proof. by rewrite -comparable_lt_minr// ereal_comparable. Qed. Lemma num_lte_minl a x y : mine x%:num y%:num < a = (x%:num < a) || (y%:num < a). Proof. by rewrite -comparable_lt_minl// ereal_comparable. Qed. End MorphReal. Section MorphGe0. Context {R : numDomainType} {nz : KnownSign.nullity}. Local Notation nR := {compare (0 : \bar R) & ?=0 & >=0}. Implicit Type x y : nR. Local Notation num := (@num _ _ (0 : \bar R) ?=0 >=0). Lemma num_abse_le a x : 0 <= a -> (`|a|%:nng <= x)%O = (a <= x%:num). Proof. by move=> a0; rewrite -num_le//= gee0_abs. Qed. Lemma num_abse_lt a x : 0 <= a -> (`|a|%:nng < x)%O = (a < x%:num). Proof. by move=> a0; rewrite -num_lt/= gee0_abs. Qed. End MorphGe0. Variant posnume_spec (R : numDomainType) (x : \bar R) : \bar R -> bool -> bool -> bool -> Type := | IsPinftyPosnume : posnume_spec x +oo false true true | IsRealPosnume (p : {posnum R}) : posnume_spec x (p%:num%:E) false true true. Lemma posnumeP (R : numDomainType) (x : \bar R) : 0 < x -> posnume_spec x x (x == 0) (0 <= x) (0 < x). Proof. case: x => [x|_|//]; last by rewrite le0y lt0y; exact: IsPinftyPosnume. rewrite lte_fin lee_fin eqe => x_gt0. rewrite x_gt0 (ltW x_gt0) (negbTE (lt0r_neq0 x_gt0)). exact: IsRealPosnume (PosNum x_gt0). Qed. Variant nonnege_spec (R : numDomainType) (x : \bar R) : \bar R -> bool -> Type := | IsPinftyNonnege : nonnege_spec x +oo true | IsRealNonnege (p : {nonneg R}) : nonnege_spec x (p%:num%:E) true. Lemma nonnegeP (R : numDomainType) (x : \bar R) : 0 <= x -> nonnege_spec x x (0 <= x). Proof. case: x => [x|_|//]; last by rewrite le0y; exact: IsPinftyNonnege. by rewrite lee_fin => /[dup] x_ge0 ->; exact: IsRealNonnege (NngNum x_ge0). Qed. Section ereal_nbhs. Context {R : numFieldType}. Local Open Scope ereal_scope. Local Open Scope classical_set_scope. Definition ereal_dnbhs (x : \bar R) (P : \bar R -> Prop) : Prop := match x with | r%:E => r^' (fun r => P r%:E) | +oo => exists M, M \is Num.real /\ forall y, M%:E < y -> P y | -oo => exists M, M \is Num.real /\ forall y, y < M%:E -> P y end. Definition ereal_nbhs (x : \bar R) (P : \bar R -> Prop) : Prop := match x with | x%:E => nbhs x (fun r => P r%:E) | +oo => exists M, M \is Num.real /\ forall y, M%:E < y -> P y | -oo => exists M, M \is Num.real /\ forall y, y < M%:E -> P y end. Canonical ereal_ereal_filter := FilteredType (extended R) (extended R) (ereal_nbhs). End ereal_nbhs. Lemma ereal_nbhs_pinfty_ge (R : numFieldType) (e : {posnum R}) : \forall x \near +oo, e%:num%:E <= x. Proof. by exists e%:num; rewrite realE ge0; split => //; apply: ltW. Qed. Lemma ereal_nbhs_ninfty_le (R : numFieldType) (r : R) : (r < 0)%R -> \forall x \near -oo, x <= r%:E. Proof. by move=> /ltW r0; exists r; rewrite realE r0 orbT; split => // x /ltW. Qed. Section ereal_nbhs_instances. Context {R : numFieldType}. Global Instance ereal_dnbhs_filter : forall x : \bar R, ProperFilter (ereal_dnbhs x). Proof. case=> [x||]. - case: (Proper_dnbhs_numFieldType x) => x0 [//= xT xI xS]. apply Build_ProperFilter' => //=; apply Build_Filter => //=. move=> P Q lP lQ; exact: xI. by move=> P Q PQ /xS; apply => y /PQ. - apply Build_ProperFilter. move=> P [x [xr xP]] //; exists (x + 1)%:E; apply xP => /=. by rewrite lte_fin ltr_addl. split=> /= [|P Q [MP [MPr gtMP]] [MQ [MQr gtMQ]] |P Q sPQ [M [Mr gtM]]]. + by exists 0%R. + have [MP0|MP0] := eqVneq MP 0%R. have [MQ0|MQ0] := eqVneq MQ 0%R. by exists 0%R; split => // x x0; split; [apply/gtMP; rewrite MP0 | apply/gtMQ; rewrite MQ0]. exists `|MQ|%R; rewrite realE normr_ge0; split => // x MQx; split. by apply: gtMP; rewrite (le_lt_trans _ MQx) // MP0 lee_fin. by apply gtMQ; rewrite (le_lt_trans _ MQx)// lee_fin real_ler_normr ?lexx. have [MQ0|MQ0] := eqVneq MQ 0%R. exists `|MP|%R; rewrite realE normr_ge0; split => // x MPx; split. by apply gtMP; rewrite (le_lt_trans _ MPx)// lee_fin real_ler_normr ?lexx. by apply gtMQ; rewrite (le_lt_trans _ MPx) // lee_fin MQ0. have {}MP0 : (0 < `|MP|)%R by rewrite normr_gt0. have {}MQ0 : (0 < `|MQ|)%R by rewrite normr_gt0. exists (Num.max (PosNum MP0) (PosNum MQ0))%:num. rewrite realE /= ge0 /=; split => //. case=> [r| |//]. * rewrite lte_fin/= num_max num_lt_maxl /= => /andP[MPx MQx]; split. by apply/gtMP; rewrite lte_fin (le_lt_trans _ MPx)// real_ler_normr ?lexx. by apply/gtMQ; rewrite lte_fin (le_lt_trans _ MQx)// real_ler_normr ?lexx. * by move=> _; split; [apply/gtMP | apply/gtMQ]. + by exists M; split => // ? /gtM /sPQ. - apply Build_ProperFilter. + move=> P [M [Mr ltMP]]; exists (M - 1)%:E. by apply: ltMP; rewrite lte_fin gtr_addl oppr_lt0. + split=> /= [|P Q [MP [MPr ltMP]] [MQ [MQr ltMQ]] |P Q sPQ [M [Mr ltM]]]. * by exists 0%R. * have [MP0|MP0] := eqVneq MP 0%R. have [MQ0|MQ0] := eqVneq MQ 0%R. by exists 0%R; split => // x x0; split; [apply/ltMP; rewrite MP0 | apply/ltMQ; rewrite MQ0]. exists (- `|MQ|)%R; rewrite realN realE normr_ge0; split => // x xMQ. split. by apply ltMP; rewrite (lt_le_trans xMQ)// lee_fin MP0 ler_oppl oppr0. apply ltMQ; rewrite (lt_le_trans xMQ) // lee_fin ler_oppl -normrN. by rewrite real_ler_normr ?realN // lexx. * have [MQ0|MQ0] := eqVneq MQ 0%R. exists (- `|MP|)%R; rewrite realN realE normr_ge0; split => // x MPx. split. apply ltMP; rewrite (lt_le_trans MPx) // lee_fin ler_oppl -normrN. by rewrite real_ler_normr ?realN // lexx. by apply ltMQ; rewrite (lt_le_trans MPx) // lee_fin MQ0 ler_oppl oppr0. have {}MP0 : (0 < `|MP|)%R by rewrite normr_gt0. have {}MQ0 : (0 < `|MQ|)%R by rewrite normr_gt0. exists (- (Num.max (PosNum MP0) (PosNum MQ0))%:num)%R. rewrite realN realE /= ge0 /=; split => //. case=> [r|//|]. - rewrite lte_fin ltr_oppr num_max num_lt_maxl => /andP[]. rewrite ltr_oppr => MPx; rewrite ltr_oppr => MQx; split. apply/ltMP; rewrite lte_fin (lt_le_trans MPx) //= ler_oppl -normrN. by rewrite real_ler_normr ?realN // lexx. apply/ltMQ; rewrite lte_fin (lt_le_trans MQx) //= ler_oppl -normrN. by rewrite real_ler_normr ?realN // lexx. - by move=> _; split; [apply/ltMP | apply/ltMQ]. * by exists M; split => // x /ltM /sPQ. Qed. Typeclasses Opaque ereal_dnbhs. Global Instance ereal_nbhs_filter : forall x, ProperFilter (@ereal_nbhs R x). Proof. case=> [r| |]. - case: (ereal_dnbhs_filter r%:E) => r0 [//= nrT rI rS]. apply: Build_ProperFilter => P /nbhs_ballP[r2 r20 rr2]. by exists r%:E; exact/rr2/ballxx. - exact: (ereal_dnbhs_filter +oo). - exact: (ereal_dnbhs_filter -oo). Qed. Typeclasses Opaque ereal_nbhs. End ereal_nbhs_instances. Section ereal_topologicalType. Variable R : realFieldType. Lemma ereal_nbhs_singleton (p : \bar R) (A : set (\bar R)) : ereal_nbhs p A -> A p. Proof. move: p => -[p | [M [Mreal MA]] | [M [Mreal MA]]] /=; [|exact: MA | exact: MA]. move=> /nbhs_ballP[_/posnumP[e]]; apply; exact/ballxx. Qed. Lemma ereal_nbhs_nbhs (p : \bar R) (A : set (\bar R)) : ereal_nbhs p A -> ereal_nbhs p (ereal_nbhs^~ A). Proof. move: p => -[p| [M [Mreal MA]] | [M [Mreal MA]]] //=. - move=> /nbhs_ballP[_/posnumP[e]] ballA. apply/nbhs_ballP; exists (e%:num / 2) => //= r per. apply/nbhs_ballP; exists (e%:num / 2) => //= x rex. apply/ballA/(@ball_splitl _ _ r) => //; exact/ball_sym. - exists (M + 1)%R; split; first by rewrite realD. move=> -[x| _ |_] //=; last by exists M. rewrite lte_fin => M'x /=. apply/nbhs_ballP; exists 1%R => //= y x1y. apply MA; rewrite lte_fin. rewrite addrC -ltr_subr_addl in M'x. rewrite (lt_le_trans M'x) // ler_subl_addl addrC -ler_subl_addl. rewrite (le_trans _ (ltW x1y)) // real_ler_norm // realB //. rewrite ltr_subr_addr in M'x. rewrite -comparabler0 (@comparabler_trans _ (M + 1)%R) //. by rewrite /Order.comparable (ltW M'x) orbT. by rewrite comparabler0 realD. by rewrite num_real. (* where we really use realFieldType *) - exists (M - 1)%R; split; first by rewrite realB. move=> -[x| _ |_] //=; last by exists M. rewrite lte_fin => M'x /=. apply/nbhs_ballP; exists 1%R => //= y x1y. apply MA; rewrite lte_fin. rewrite ltr_subr_addl in M'x. rewrite (le_lt_trans _ M'x) // addrC -ler_subl_addl. rewrite (le_trans _ (ltW x1y)) // distrC real_ler_norm // realB //. by rewrite num_real. (* where we really use realFieldType *) rewrite addrC -ltr_subr_addr in M'x. rewrite -comparabler0 (@comparabler_trans _ (M - 1)%R) //. by rewrite /Order.comparable (ltW M'x). by rewrite comparabler0 realB. Qed. Definition ereal_topologicalMixin : Topological.mixin_of (@ereal_nbhs R) := topologyOfFilterMixin _ ereal_nbhs_singleton ereal_nbhs_nbhs. Canonical ereal_topologicalType := TopologicalType _ ereal_topologicalMixin. End ereal_topologicalType. Local Open Scope classical_set_scope. Lemma nbhsNe (R : realFieldType) (x : \bar R) : nbhs (- x) = [set (-%E @` A) | A in nbhs x]. Proof. case: x => [r /=| |]. - rewrite /nbhs /= /ereal_nbhs -nbhs_ballE. rewrite predeqE => S; split => [[_/posnumP[e] reS]|[S' [_ /posnumP[e] reS' <-]]]. exists (-%E @` S). exists e%:num => //= r1 rer1; exists (- r1%:E); last by rewrite oppeK. by apply reS; rewrite /ball /= opprK -normrN opprD opprK. rewrite predeqE => s; split => [[y [z Sz] <- <-]|Ss]. by rewrite oppeK. by exists (- s); [exists s | rewrite oppeK]. exists e%:num => //= r1 rer1; exists (- r1%:E); last by rewrite oppeK. by apply reS'; rewrite /ball /= opprK -normrN opprD. - rewrite predeqE => S; split=> [[M [Mreal MS]]|[x [M [Mreal Mx]] <-]]. exists (-%E @` S). exists (- M)%R; rewrite realN Mreal; split => // x Mx. by exists (- x); [apply MS; rewrite lte_oppl | rewrite oppeK]. rewrite predeqE => x; split=> [[y [z Sz <- <-]]|Sx]; first by rewrite oppeK. by exists (- x); [exists x | rewrite oppeK]. exists (- M)%R; rewrite realN; split => // y yM. exists (- y); by [apply Mx; rewrite lte_oppr|rewrite oppeK]. - rewrite predeqE => S; split=> [[M [Mreal MS]]|[x [M [Mreal Mx]] <-]]. exists (-%E @` S). exists (- M)%R; rewrite realN Mreal; split => // x Mx. by exists (- x); [apply MS; rewrite lte_oppr | rewrite oppeK]. rewrite predeqE => x; split=> [[y [z Sz <- <-]]|Sx]; first by rewrite oppeK. by exists (- x); [exists x | rewrite oppeK]. exists (- M)%R; rewrite realN; split => // y yM. exists (- y); by [apply Mx; rewrite lte_oppl|rewrite oppeK]. Qed. Lemma nbhsNKe (R : realFieldType) (z : \bar R) (A : set (\bar R)) : nbhs (- z) (-%E @` A) -> nbhs z A. Proof. rewrite nbhsNe => -[S zS] SA; rewrite -(oppeK z) nbhsNe. exists (-%E @` S); first by rewrite nbhsNe; exists S. rewrite predeqE => x; split => [[y [u Su <-{y} <-]]|Ax]. rewrite oppeK. move: SA; rewrite predeqE => /(_ (- u)) [h _]. have : (exists2 y, S y & - y = - u) by exists u. by move/h => -[y Ay] /eqP; rewrite eqe_opp => /eqP <-. exists (- x); last by rewrite oppeK. exists x => //. move: SA; rewrite predeqE => /(_ (- x)) [_ h]. have : (-%E @` A) (- x) by exists x. by move/h => [y Sy] /eqP; rewrite eqe_opp => /eqP <-. Qed. Lemma oppe_continuous (R : realFieldType) : continuous (@oppe R). Proof. move=> x S /= xS; apply nbhsNKe; rewrite image_preimage //. by rewrite predeqE => y; split => // _; exists (- y) => //; rewrite oppeK. Qed. Section contract_expand. Variable R : realFieldType. Implicit Types (x : \bar R) (r : R). Local Open Scope ereal_scope. Definition contract x : R := match x with | r%:E => r / (1 + `|r|) | +oo => 1 | -oo => -1 end. Lemma contract_lt1 r : (`|contract r%:E| < 1)%R. Proof. rewrite normrM normrV ?unitfE //. rewrite ltr_pdivr_mulr // ?mul1r//; last by rewrite gtr0_norm. by rewrite [ltRHS]gtr0_norm ?ltr_addr// ltr_spaddl. Qed. Lemma contract_le1 x : (`|contract x| <= 1)%R. Proof. by case: x => [r| |] /=; rewrite ?normrN1 ?normr1 // (ltW (contract_lt1 _)). Qed. Lemma contract0 : contract 0 = 0%R. Proof. by rewrite /contract mul0r. Qed. Lemma contractN x : contract (- x) = (- contract x)%R. Proof. by case: x => //= [r|]; [ rewrite normrN mulNr | rewrite opprK]. Qed. Lemma contract_imageN (S : set (\bar R)) : contract @` (-%E @` S) = -%R @` (contract @` S). Proof. rewrite predeqE => r; split => [[y [z Sz <-{y} <-{r}]]|[s [y Sy <-{s} <-{r}]]]. by exists (contract z); [exists z | rewrite contractN]. by exists (- y); [exists y | rewrite contractN]. Qed. (* TODO: not exploited yet: expand is nondecreasing everywhere so it should be possible to use some of the homoRL/homoLR lemma where monoRL/monoLR do not apply *) Definition expand r : \bar R := if (r >= 1)%R then +oo else if (r <= -1)%R then -oo else (r / (1 - `|r|))%:E. Lemma expand1 r : (1 <= r)%R -> expand r = +oo. Proof. by move=> r1; rewrite /expand r1. Qed. Lemma expandN r : expand (- r)%R = - expand r. Proof. rewrite /expand; case: ifPn => [r1|]. rewrite ifF; [by rewrite ifT // -ler_oppr|apply/negbTE]. by rewrite -ltNge -(opprK r) -ltr_oppl (lt_le_trans _ r1) // -subr_gt0 opprK. rewrite -ltNge => r1; case: ifPn; rewrite ler_oppl opprK; [by move=> ->|]. by rewrite -ltNge leNgt => ->; rewrite leNgt -ltr_oppl r1 /= mulNr normrN. Qed. Lemma expandN1 r : (r <= -1)%R -> expand r = -oo. Proof. by rewrite ler_oppr => /expand1/eqP; rewrite expandN eqe_oppLR => /eqP. Qed. Lemma expand0 : expand 0%R = 0. Proof. by rewrite /expand leNgt ltr01 /= oppr_ge0 leNgt ltr01 /= mul0r. Qed. Lemma expandK : {in [pred r | `|r| <= 1]%R, cancel expand contract}. Proof. move=> r; rewrite inE le_eqVlt => /orP[|r1]. rewrite eqr_norml => /andP[/orP[]/eqP->{r}] _; by [rewrite expand1|rewrite expandN1]. rewrite /expand 2!leNgt ltr_oppl; case/ltr_normlP : (r1) => -> -> /=. have r_pneq0 : (1 + r / (1 - r) != 0)%R. rewrite -[X in (X + _)%R](@divrr _ (1 - r)%R) -?mulrDl; last first. by rewrite unitfE subr_eq0 eq_sym lt_eqF // ltr_normlW. by rewrite subrK mulf_neq0 // invr_eq0 subr_eq0 eq_sym lt_eqF // ltr_normlW. have r_nneq0 : (1 - r / (1 + r) != 0)%R. rewrite -[X in (X + _)%R](@divrr _ (1 + r)%R) -?mulrBl; last first. by rewrite unitfE addrC addr_eq0 gt_eqF // ltrNnormlW. rewrite addrK mulf_neq0 // invr_eq0 addr_eq0 -eqr_oppLR eq_sym gt_eqF //. exact: ltrNnormlW. wlog : r r1 r_pneq0 r_nneq0 / (0 <= r)%R => wlog_r0. have [r0|r0] := lerP 0 r; first by rewrite wlog_r0. move: (wlog_r0 (- r)%R). rewrite !(normrN, opprK, mulNr) oppr_ge0 => /(_ r1 r_nneq0 r_pneq0 (ltW r0)). by move/eqP; rewrite eqr_opp => /eqP. rewrite /contract !ger0_norm //; last first. by rewrite divr_ge0 // subr_ge0 (le_trans _ (ltW r1)) // ler_norm. apply: (@mulIr _ (1 + r / (1 - r))%R); first by rewrite unitfE. rewrite -(mulrA (r / _)) mulVr ?unitfE // mulr1. rewrite -[X in (X + _ / _)%R](@divrr _ (1 - r)%R) -?mulrDl ?subrK ?div1r //. by rewrite unitfE subr_eq0 eq_sym lt_eqF // ltr_normlW. Qed. Lemma le_contract : {mono contract : x y / (x <= y)%O}. Proof. apply: le_mono; move=> -[r0 | | ] [r1 | _ | _] //=. - rewrite lte_fin => r0r1; rewrite ltr_pdivr_mulr ?ltr_paddr//. rewrite mulrAC ltr_pdivl_mulr ?ltr_paddr// 2?mulrDr 2?mulr1. have [r10|?] := ler0P r1; last first. rewrite ltr_le_add // mulrC; have [r00|//] := ler0P r0. by rewrite (@le_trans _ _ 0%R) // ?pmulr_lle0// mulr_ge0// ?oppr_ge0// ltW. have [?|r00] := ler0P r0; first by rewrite ltr_le_add // 2!mulrN mulrC. by move: (le_lt_trans r10 (lt_trans r00 r0r1)); rewrite ltxx. - by rewrite ltr_pdivr_mulr ?ltr_paddr// mul1r ltr_spaddl // ler_norm. - rewrite ltr_pdivl_mulr ?mulN1r ?ltr_paddr// => _. by rewrite ltr_oppl ltr_spaddl // ler_normr lexx orbT. - by rewrite -subr_gt0 opprK. Qed. Definition lt_contract := leW_mono le_contract. Definition contract_inj := mono_inj lexx le_anti le_contract. Lemma le_expand_in : {in [pred r | `|r| <= 1]%R &, {mono expand : x y / (x <= y)%O}}. Proof. exact: can_mono_in (onW_can_in predT expandK) _ (in2W le_contract). Qed. Definition lt_expand := leW_mono_in le_expand_in. Definition expand_inj := mono_inj_in lexx le_anti le_expand_in. Lemma fine_expand r : (`|r| < 1)%R -> (fine (expand r))%:E = expand r. Proof. by move=> r1; rewrite /expand 2!leNgt ltr_oppl; case/ltr_normlP : r1 => -> ->. Qed. Lemma contractK : cancel contract expand. Proof. apply: (onS_can [pred r | `|r| <= 1]%R contract_le1). exact: inj_can_sym_on expandK (on2W contract_inj). Qed. Lemma bijective_contract : {on [pred r | `|r| <= 1]%R, bijective contract}. Proof. exists expand; [exact: in1W contractK | exact: expandK]. Qed. Definition le_expandLR := monoLR_in (in_onW_can _ predT contractK) (fun x _ => contract_le1 x) le_expand_in. Definition lt_expandLR := monoLR_in (in_onW_can _ predT contractK) (fun x _ => contract_le1 x) lt_expand. Definition le_expandRL := monoRL_in (in_onW_can _ predT contractK) (fun x _ => contract_le1 x) le_expand_in. Definition lt_expandRL := monoRL_in (in_onW_can _ predT contractK) (fun x _ => contract_le1 x) lt_expand. Lemma le_expand : {homo expand : x y / (x <= y)%O}. Proof. move=> x y xy; have [x1|] := lerP `|x| 1. have [y_le1|/ltW /expand1->] := leP y 1%R; last by rewrite leey. rewrite le_expand_in ?inE// ler_norml y_le1 (le_trans _ xy)//. by rewrite ler_oppl (ler_normlP _ _ _). rewrite ltr_normr => /orP[|] x1; last first. by rewrite expandN1 // ?leNye // ler_oppr ltW. by rewrite expand1; [rewrite expand1 // (le_trans _ xy) // ltW | exact: ltW]. Qed. Lemma contract_eq0 x : (contract x == 0%R) = (x == 0). Proof. by rewrite -(can_eq contractK) contract0. Qed. Lemma contract_eqN1 x : (contract x == -1) = (x == -oo). Proof. by rewrite -(can_eq contractK). Qed. Lemma contract_eq1 x : (contract x == 1%R) = (x == +oo). Proof. by rewrite -(can_eq contractK). Qed. Lemma expand_eqoo r : (expand r == +oo) = (1 <= r)%R. Proof. by rewrite /expand; case: ifP => //; case: ifP. Qed. Lemma expand_eqNoo r : (expand r == -oo) = (r <= -1)%R. Proof. rewrite /expand; case: ifP => /= r1; last by case: ifP. by apply/esym/negbTE; rewrite -ltNge (lt_le_trans _ r1) // -subr_gt0 opprK. Qed. End contract_expand. Section contract_expand_realType. Variable R : realType. Let contract := @contract R. Lemma sup_contract_le1 S : S !=set0 -> (`|sup (contract @` S)| <= 1)%R. Proof. case=> x Sx; rewrite ler_norml; apply/andP; split; last first. apply sup_le_ub; first by exists (contract x), x. by move=> r [y Sy] <-; case/ler_normlP : (contract_le1 y). rewrite (@le_trans _ _ (contract x)) //. by case/ler_normlP : (contract_le1 x); rewrite ler_oppl. apply sup_ub; last by exists x. by exists 1%R => r [y Sy <-]; case/ler_normlP : (contract_le1 y). Qed. Lemma contract_sup S : S !=set0 -> contract (ereal_sup S) = sup (contract @` S). Proof. move=> S0; apply/eqP; rewrite eq_le; apply/andP; split; last first. apply sup_le_ub. by case: S0 => x Sx; exists (contract x), x. move=> x [y Sy] <-{x}; rewrite le_contract; exact/ereal_sup_ub. rewrite leNgt; apply/negP. set supc := sup _; set csup := contract _; move=> ltsup. suff [y [ysupS ?]] : exists y, y < ereal_sup S /\ ubound S y. have : ereal_sup S <= y by apply ub_ereal_sup. by move/(lt_le_trans ysupS); rewrite ltxx. suff [x [? [ubSx x1]]] : exists x, (x < csup)%R /\ ubound (contract @` S) x /\ (`|x| <= 1)%R. exists (expand x); split => [|y Sy]. by rewrite -(contractK (ereal_sup S)) lt_expand // inE // contract_le1. by rewrite -(contractK y) le_expand //; apply ubSx; exists y. exists ((supc + csup) / 2); split; first by rewrite midf_lt. split => [r [y Sy <-{r}]|]. rewrite (@le_trans _ _ supc) ?midf_le //; last by rewrite ltW. apply sup_ub; last by exists y. by exists 1%R => r [z Sz <-]; case/ler_normlP : (contract_le1 z). rewrite ler_norml; apply/andP; split; last first. rewrite ler_pdivr_mulr // mul1r (_ : 2 = 1 + 1)%R // ler_add //. by case/ler_normlP : (sup_contract_le1 S0). by case/ler_normlP : (contract_le1 (ereal_sup S)). rewrite ler_pdivl_mulr // (_ : 2 = 1 + 1)%R // mulN1r opprD ler_add //. by case/ler_normlP : (sup_contract_le1 S0); rewrite ler_oppl. by case/ler_normlP : (contract_le1 (ereal_sup S)); rewrite ler_oppl. Qed. Lemma contract_inf S : S !=set0 -> contract (ereal_inf S) = inf (contract @` S). Proof. move=> -[x Sx]; rewrite /ereal_inf /contract (contractN (ereal_sup (-%E @` S))). by rewrite -/contract contract_sup /inf; [rewrite contract_imageN | exists (- x), x]. Qed. End contract_expand_realType. Section ereal_PseudoMetric. Variable R : realFieldType. Implicit Types (x y : \bar R) (r : R). Definition ereal_ball x r y := (`|contract x - contract y| < r)%R. Lemma ereal_ball_center x r : (0 < r)%R -> ereal_ball x r x. Proof. by move=> e0; rewrite /ereal_ball subrr normr0. Qed. Lemma ereal_ball_sym x y r : ereal_ball x r y -> ereal_ball y r x. Proof. by rewrite /ereal_ball distrC. Qed. Lemma ereal_ball_triangle x y z r1 r2 : ereal_ball x r1 y -> ereal_ball y r2 z -> ereal_ball x (r1 + r2) z. Proof. rewrite /ereal_ball => h1 h2; rewrite -[X in (X - _)%R](subrK (contract y)). by rewrite -addrA (le_lt_trans (ler_norm_add _ _)) // ltr_add. Qed. Lemma ereal_ballN x y (e : {posnum R}) : ereal_ball (- x) e%:num (- y) -> ereal_ball x e%:num y. Proof. by rewrite /ereal_ball 2!contractN opprK -opprB normrN addrC. Qed. Lemma le_ereal_ball x : {homo ereal_ball x : e e' / (e <= e')%R >-> e `<=` e'}. Proof. by move=> e e' ee' y; rewrite /ereal_ball => /lt_le_trans; exact. Qed. Lemma ereal_ball_ninfty_oversize (e : {posnum R}) x : (2 < e%:num)%R -> ereal_ball -oo e%:num x. Proof. move=> e2; rewrite /ereal_ball /= (le_lt_trans _ e2) // -opprB normrN opprK. rewrite (le_trans (ler_norm_add _ _)) // normr1 -ler_subr_addr. by rewrite (le_trans (contract_le1 _)) // (_ : 2 = 1 + 1)%R // addrK. Qed. Lemma contract_ereal_ball_pinfty r (e : {posnum R}) : (1 < contract r%:E + e%:num)%R -> ereal_ball r%:E e%:num +oo. Proof. move=> re1; rewrite /ereal_ball; rewrite [contract +oo]/= ler0_norm; last first. by rewrite subr_le0; case/ler_normlP: (contract_le1 r%:E). by rewrite opprB ltr_subl_addl. Qed. Lemma expand_ereal_ball_pinfty {e : {posnum R}} r : (e%:num <= 1)%R -> expand (1 - e%:num)%R < r%:E -> ereal_ball +oo e%:num r%:E. Proof. move=> e1 er; rewrite /ereal_ball gtr0_norm ?subr_gt0; last first. by case/ltr_normlP : (contract_lt1 r). rewrite ltr_subl_addl addrC -ltr_subl_addl -[ltLHS]expandK ?lt_contract//. by rewrite inE ger0_norm ?ler_subl_addl ?ler_addr // subr_ge0. Qed. Lemma contract_ereal_ball_fin_le r r' (e : {posnum R}) : (r <= r')%R -> (1 <= contract r%:E + e%:num)%R -> ereal_ball r%:E e%:num r'%:E. Proof. rewrite le_eqVlt => /predU1P[<-{r'} _|rr' re1]; first exact: ereal_ball_center. rewrite /ereal_ball ltr0_norm; last by rewrite subr_lt0 lt_contract lte_fin. rewrite opprB ltr_subl_addl (lt_le_trans _ re1) //. by case/ltr_normlP : (contract_lt1 r'). Qed. Lemma contract_ereal_ball_fin_lt r r' (e : {posnum R}) : (r' < r)%R -> (contract r%:E - e%:num <= -1)%R -> ereal_ball r%:E e%:num r'%:E. Proof. move=> r'r reN1; rewrite /ereal_ball. rewrite gtr0_norm ?subr_gt0 ?lt_contract ?lte_fin//. rewrite ltr_subl_addl addrC -ltr_subl_addl (le_lt_trans reN1) //. by move: (contract_lt1 r'); rewrite ltr_norml => /andP[]. Qed. Lemma expand_ereal_ball_fin_lt r' r (e : {posnum R}) : (r' < r)%R -> expand (contract r%:E - e%:num)%R < r'%:E -> (`|contract r%:E - e%:num| < 1)%R -> ereal_ball r%:E e%:num r'%:E. Proof. move=> r'r ? r'e'r. rewrite /ereal_ball gtr0_norm ?subr_gt0 ?lt_contract ?lte_fin//. by rewrite ltr_subl_addl addrC -ltr_subl_addl -lt_expandLR ?inE ?ltW. Qed. Lemma ball_ereal_ball_fin_lt r r' (e : {posnum R}) : let e' := (r - fine (expand (contract r%:E - e%:num)))%R in ball r e' r' -> (r' < r)%R -> (`|contract r%:E - (e)%:num| < 1)%R -> ereal_ball r%:E (e)%:num r'%:E. Proof. move=> e' re'r' rr' X; rewrite /ereal_ball. rewrite gtr0_norm ?subr_gt0// ?lt_contract ?lte_fin//. move: re'r'. rewrite /ball /= gtr0_norm // ?subr_gt0// /e'. rewrite -ltr_subl_addl addrAC subrr add0r ltr_oppl opprK -lte_fin. rewrite fine_expand // lt_expandLR ?inE ?ltW//. by rewrite ltr_subl_addl addrC -ltr_subl_addl. Qed. Lemma ball_ereal_ball_fin_le r r' (e : {posnum R}) : let e' : R := (fine (expand (contract r%:E + e%:num)) - r)%R in ball r e' r' -> (r <= r')%R -> (`| contract r%:E + e%:num | < 1)%R -> (ereal_ball r%:E e%:num r'%:E). Proof. move=> e' r'e'r rr' re1; rewrite /ereal_ball. move: rr'; rewrite le_eqVlt => /predU1P[->|rr']; first by rewrite subrr normr0. rewrite /ball /= ltr0_norm ?subr_lt0// opprB in r'e'r. rewrite ltr0_norm ?subr_lt0 ?lt_contract ?lte_fin//. rewrite opprB; move: r'e'r. rewrite /e' -ltr_subl_addr opprK subrK -lte_fin fine_expand //. by rewrite lt_expandRL ?inE ?ltW// ltr_subl_addl. Qed. Lemma nbhs_oo_up_e1 (A : set (\bar R)) (e : {posnum R}) : (e%:num <= 1)%R -> ereal_ball +oo e%:num `<=` A -> nbhs +oo A. Proof. move=> e1 ooeA. exists (fine (expand (1 - e%:num)%R)); rewrite num_real; split => //. case => [r | | //]. - rewrite fine_expand; last first. by rewrite ger0_norm ?ltr_subl_addl ?ltr_addr // subr_ge0. by move=> ?; exact/ooeA/expand_ereal_ball_pinfty. - by move=> _; exact/ooeA/ereal_ball_center. Qed. Lemma nbhs_oo_down_e1 (A : set (\bar R)) (e : {posnum R}) : (e%:num <= 1)%R -> ereal_ball -oo e%:num `<=` A -> nbhs -oo A. Proof. move=> e1 reA; suff h : nbhs +oo (-%E @` A). rewrite (_ : -oo = - +oo) // nbhsNe; exists (-%E @` A) => //. rewrite predeqE => x; split=> [[y [z Az <- <-]]|Ax]; rewrite ?oppeK //. by exists (- x); [exists x | rewrite oppeK]. apply (@nbhs_oo_up_e1 _ e) => // x x1e; exists (- x); last by rewrite oppeK. by apply/reA/ereal_ballN; rewrite oppeK. Qed. Lemma nbhs_oo_up_1e (A : set (\bar R)) (e : {posnum R}) : (1 < e%:num)%R -> ereal_ball +oo e%:num `<=` A -> nbhs +oo A. Proof. move=> e1 reA; have [e2{e1}|e2] := ltrP 2 e%:num. suff -> : A = setT by exists 0%R. rewrite predeqE => x; split => // _; apply reA. exact/ereal_ballN/ereal_ball_ninfty_oversize. have /andP[e10 e11] : (0 < e%:num - 1 <= 1)%R. by rewrite subr_gt0 e1 /= ler_subl_addl. apply nbhsNKe. have : ((PosNum e10)%:num <= 1)%R by []. move/(@nbhs_oo_down_e1 (-%E @` A) (PosNum e10)); apply. move=> y ye; exists (- y); last by rewrite oppeK. apply/reA/ereal_ballN; rewrite oppeK /=. by apply: le_ereal_ball ye => /=; rewrite ler_subl_addl ler_addr. Qed. Lemma nbhs_oo_down_1e (A : set (\bar R)) (e : {posnum R}) : (1 < e%:num)%R -> ereal_ball -oo e%:num `<=` A -> nbhs -oo A. Proof. move=> e1 reA; have [e2{e1}|e2] := ltrP 2 e%:num. suff -> : A = setT by exists 0%R. by rewrite predeqE => x; split => // _; exact/reA/ereal_ball_ninfty_oversize. have /andP[e10 e11] : (0 < e%:num - 1 <= 1)%R. by rewrite subr_gt0 e1 /= ler_subl_addl. apply nbhsNKe. have : ((PosNum e10)%:num <= 1)%R by []. move/(@nbhs_oo_up_e1 (-%E @` A) (PosNum e10)); apply. move=> y ye; exists (- y); last by rewrite oppeK. apply/reA/ereal_ballN; rewrite /= oppeK. by apply: le_ereal_ball ye => /=; rewrite ler_subl_addl ler_addr. Qed. Lemma nbhs_fin_out_above r (e : {posnum R}) (A : set (\bar R)) : ereal_ball r%:E e%:num `<=` A -> (- 1 < contract r%:E - e%:num)%R -> (1 <= contract r%:E + e%:num)%R -> nbhs r%:E A. Proof. move=> reA reN1 re1. have er1 : (`|contract r%:E - e%:num| < 1)%R. rewrite ltr_norml reN1 andTb ltr_subl_addl ltr_spaddl //. by move: (contract_le1 r%:E); rewrite ler_norml => /andP[]. pose e' := (r - fine (expand (contract r%:E - e%:num)))%R. have e'0 : (0 < e')%R. rewrite subr_gt0 -lte_fin -[ltRHS](contractK r%:E). rewrite fine_expand // lt_expand ?inE ?contract_le1// ?ltW//. by rewrite ltr_subl_addl ltr_addr. apply/nbhs_ballP; exists e' => // r' re'r'; apply reA. by have [?|?] := lerP r r'; [exact: contract_ereal_ball_fin_le | exact: ball_ereal_ball_fin_lt]. Qed. Lemma nbhs_fin_out_below r (e : {posnum R}) (A : set (\bar R)) : ereal_ball r%:E e%:num `<=` A -> (contract r%:E - e%:num <= - 1)%R -> (contract r%:E + e%:num < 1)%R -> nbhs r%:E A. Proof. move=> reA reN1 re1. have ? : (`|contract r%:E + e%:num| < 1)%R. rewrite ltr_norml re1 andbT (@lt_le_trans _ _ (contract r%:E)) // ?ler_addl //. by move: (contract_lt1 r); rewrite ltr_norml => /andP[]. pose e' : R := (fine (expand (contract r%:E + e%:num)) - r)%R. have e'0 : (0 < e')%R. rewrite /e' subr_gt0 -lte_fin -[in ltLHS](contractK r%:E). by rewrite fine_expand // lt_expand ?inE ?contract_le1 ?ltr_addl ?ltW. apply/nbhs_ballP; exists e' => // r' r'e'r; apply reA. by have [?|?] := lerP r r'; [exact: ball_ereal_ball_fin_le | exact: contract_ereal_ball_fin_lt]. Qed. Lemma nbhs_fin_out_above_below r (e : {posnum R}) (A : set (\bar R)) : ereal_ball r%:E e%:num `<=` A -> (contract r%:E - e%:num < -1)%R -> (1 < contract r%:E + e%:num)%R -> nbhs r%:E A. Proof. move=> reA reN1 re1; suff : A = setT by move->; apply: filterT. rewrite predeqE => x; split => // _; apply reA. case: x => [r'| |] //. - have [?|?] := lerP r r'. + by apply: contract_ereal_ball_fin_le => //; exact/ltW. + by apply contract_ereal_ball_fin_lt => //; exact/ltW. - exact/contract_ereal_ball_pinfty. - apply/ereal_ballN/contract_ereal_ball_pinfty. by rewrite EFinN contractN -(opprK 1%R) ltr_oppl opprD opprK. Qed. Lemma nbhs_fin_inbound r (e : {posnum R}) (A : set (\bar R)) : ereal_ball r%:E e%:num `<=` A -> nbhs r%:E A. Proof. move=> reA. have [|reN1] := boolP (contract r%:E - e%:num == -1)%R. rewrite subr_eq addrC => /eqP reN1. have [re1|] := boolP (contract r%:E + e%:num == 1)%R. move/eqP : reN1; rewrite -(eqP re1) opprD addrCA subrr addr0 -subr_eq0. rewrite opprK -mulr2n mulrn_eq0 orFb contract_eq0 => /eqP[r0]. move: re1; rewrite r0 contract0 add0r => /eqP e1. apply/nbhs_ballP; exists 1%R => //= r'; rewrite /ball /= sub0r normrN => r'1. apply reA. by rewrite /ereal_ball r0 contract0 sub0r normrN e1 contract_lt1. rewrite neq_lt => /orP[re1|re1]. by apply (@nbhs_fin_out_below _ e) => //; rewrite reN1 addrAC subrr sub0r. have e1 : (1 < e%:num)%R. move: re1; rewrite reN1 addrAC ltr_subr_addl -!mulr2n -(mulr_natl e%:num). by rewrite -{1}(mulr1 2) => ?; rewrite -(@ltr_pmul2l _ 2). have Aoo : setT `\ -oo `<=` A. move=> x [_]; rewrite /set1 /= => xnoo; apply reA. case: x xnoo => [r' _ | _ |//]. have [rr'|r'r] := lerP (contract r%:E) (contract r'%:E). apply: contract_ereal_ball_fin_le; last exact/ltW. by rewrite -lee_fin -(contractK r%:E) -(contractK r'%:E) le_expand. apply: contract_ereal_ball_fin_lt; last first. by rewrite reN1 addrAC subrr add0r. rewrite -lte_fin -(contractK r%:E) -(contractK r'%:E). by rewrite lt_expand // inE; exact: contract_le1. exact: contract_ereal_ball_pinfty. have : nbhs r%:E (setT `\ -oo) by apply/nbhs_ballP; exists 1%R => /=. move=> /nbhs_ballP[_/posnumP[e']] /=; rewrite /ball /= => h. by apply/nbhs_ballP; exists e'%:num => //= y /h; apply: Aoo. move: reN1; rewrite eq_sym neq_lt => /orP[reN1|reN1]. have [re1|re1] := eqVneq (contract r%:E + e%:num)%R 1%R. by apply (@nbhs_fin_out_above _ e) => //; rewrite re1. move: re1; rewrite neq_lt => /orP[re1|re1]. have ? : (`|contract r%:E - e%:num| < 1)%R. rewrite ltr_norml reN1 andTb ltr_subl_addl. rewrite (@lt_le_trans _ _ 1%R) // ?ler_addr//. by case/ltr_normlP : (contract_lt1 r). have ? : (`|contract r%:E + e%:num| < 1)%R. rewrite ltr_norml re1 andbT -(addr0 (-1)) ler_lt_add //. by move: (contract_le1 r%:E); rewrite ler_norml => /andP[]. pose e' : R := Num.min (r - fine (expand (contract r%:E - e%:num)))%R (fine (expand (contract r%:E + e%:num)) - r)%R. have e'0 : (0 < e')%R. rewrite /e' lt_minr; apply/andP; split. rewrite subr_gt0 -lte_fin -[in ltRHS](contractK r%:E). rewrite fine_expand // lt_expand// ?inE ?contract_le1 ?ltW//. by rewrite ltr_subl_addl ltr_addr. rewrite subr_gt0 -lte_fin -[in ltLHS](contractK r%:E). by rewrite fine_expand// lt_expand ?inE ?contract_le1 ?ltr_addl ?ltW. apply/nbhs_ballP; exists e' => // r' re'r'; apply reA. have [|r'r] := lerP r r'. move=> rr'; apply: ball_ereal_ball_fin_le => //. by apply: le_ball re'r'; rewrite le_minl lexx orbT. move: re'r'; rewrite /ball /= lt_minr => /andP[]. rewrite gtr0_norm ?subr_gt0 // -ltr_subl_addl addrAC subrr add0r ltr_oppl. rewrite opprK -lte_fin fine_expand // => r'e'r _. exact: expand_ereal_ball_fin_lt. by apply (@nbhs_fin_out_above _ e) => //; rewrite ltW. have [re1|re1] := ltrP 1 (contract r%:E + e%:num). exact: (@nbhs_fin_out_above_below _ e). move: re1; rewrite le_eqVlt => /orP[re1|re1]. have {}re1 : contract r%:E = (1 - e%:num)%R. by move: re1; rewrite eq_sym -subr_eq => /eqP <-. have e1 : (1 < e%:num)%R. move: reN1. rewrite re1 -addrA -opprD ltr_subl_addl ltr_subr_addl -!mulr2n. rewrite -(mulr_natl e%:num) -{1}(mulr1 2) => ?. by rewrite -(@ltr_pmul2l _ 2). have Aoo : (setT `\ +oo `<=` A). move=> x [_]; rewrite /set1 /= => xpoo; apply reA. case: x xpoo => [r' _ | // |_]. rewrite /ereal_ball. have [rr'|r'r] := lerP (contract r%:E) (contract r'%:E). rewrite re1 opprB addrCA -[ltRHS]addr0 ltr_add2 subr_lt0. by case/ltr_normlP : (contract_lt1 r'). rewrite /ereal_ball. rewrite re1 addrAC ltr_subl_addl ltr_add // (lt_trans _ e1) // ltr_oppl. by move: (contract_lt1 r'); rewrite ltr_norml => /andP[]. rewrite /ereal_ball. rewrite [contract -oo]/= opprK gtr0_norm ?subr_gt0; last first. rewrite -ltr_subl_addl add0r ltr_oppl. by move: (contract_lt1 r); rewrite ltr_norml => /andP[]. by rewrite re1 addrAC ltr_subl_addl ltr_add. have : nbhs r%:E (setT `\ +oo) by exists 1%R => /=. case => _/posnumP[x] /=; rewrite /ball_ => h. by exists x%:num => //= y /h; exact: Aoo. by apply (@nbhs_fin_out_below _ e) => //; rewrite ltW. Qed. Lemma ereal_nbhsE : nbhs = nbhs_ (entourage_ ereal_ball). Proof. set diag := fun r => [set xy | ereal_ball xy.1 r xy.2]. rewrite predeq2E => x A; split. - rewrite {1}/nbhs /= /ereal_nbhs. case: x => [/= r [_/posnumP[e] reA]| [M [/= Mreal MA]]| [M [/= Mreal MA]]]. + pose e' : R := Num.min (contract r%:E - contract (r%:E - e%:num%:E))%R (contract (r%:E + e%:num%:E) - contract r%:E)%R. exists (diag e'); rewrite /diag. exists e' => //. rewrite /= /e' lt_minr; apply/andP; split. by rewrite subr_gt0 lt_contract lte_fin ltr_subl_addr ltr_addl. by rewrite subr_gt0 lt_contract lte_fin ltr_addl. case=> [r' /= re'r'| |]/=. * rewrite /ereal_ball in re'r'. have [r'r|rr'] := lerP (contract r'%:E) (contract r%:E). apply: reA; rewrite /ball /= real_ltr_norml // ?num_real //. rewrite ger0_norm ?subr_ge0// in re'r'. have : (contract (r%:E - e%:num%:E) < contract r'%:E)%R. move: re'r'; rewrite /e' lt_minr => /andP[+ _]. rewrite /e' ltr_subr_addl addrC -ltr_subr_addl => /lt_le_trans. by apply; rewrite opprB addrCA subrr addr0. rewrite -lt_expandRL ?inE ?contract_le1 // !contractK lte_fin. rewrite ltr_subl_addr addrC -ltr_subl_addr => ->; rewrite andbT. rewrite (@lt_le_trans _ _ 0%R)// 1?ltr_oppl 1?oppr0// subr_ge0. by rewrite -lee_fin -le_contract. apply: reA; rewrite /ball /= real_ltr_norml // ?num_real //. rewrite ltr0_norm ?subr_lt0// opprB in re'r'. apply/andP; split; last first. by rewrite (@lt_trans _ _ 0%R) // subr_lt0 -lte_fin -lt_contract. rewrite ltr_oppl opprB. rewrite /e' in re'r'. have r're : (contract r'%:E < contract (r%:E + e%:num%:E))%R. move: re'r'; rewrite lt_minr => /andP[_]. by rewrite ltr_subl_addr subrK. rewrite ltr_subl_addr -lte_fin -(contractK (_ + r)%:E)%R. by rewrite addrC -(contractK r'%:E) // lt_expand ?inE ?contract_le1. * rewrite /ereal_ball [contract +oo]/=. rewrite lt_minr => /andP[re'1 re'2]. have [cr0|cr0] := lerP 0 (contract r%:E). move: re'2; rewrite ler0_norm; last first. by rewrite subr_le0; case/ler_normlP : (contract_le1 r%:E). rewrite opprB ltr_subr_addl addrCA subrr addr0 => h. exfalso. move: h; apply/negP; rewrite -leNgt. by case/ler_normlP : (contract_le1 (r%:E + e%:num%:E)). move: re'2; rewrite ler0_norm; last first. by rewrite subr_le0; case/ler_normlP : (contract_le1 r%:E). rewrite opprB ltr_subr_addl addrCA subrr addr0 => h. exfalso. move: h; apply/negP; rewrite -leNgt. by case/ler_normlP : (contract_le1 (r%:E + e%:num%:E)). * rewrite /ereal_ball [contract -oo]/=; rewrite opprK. rewrite lt_minr => /andP[re'1 _]. move: re'1. rewrite ger0_norm; last first. rewrite addrC -ler_subl_addl add0r. by move: (contract_le1 r%:E); rewrite ler_norml => /andP[]. rewrite ltr_add2l => h. exfalso. move: h; apply/negP; rewrite -leNgt -ler_oppl. by move: (contract_le1 (r%:E - e%:num%:E)); rewrite ler_norml => /andP[]. + exists (diag (1 - contract M%:E))%R; rewrite /diag. exists (1 - contract M%:E)%R => //=. by rewrite subr_gt0 (le_lt_trans _ (contract_lt1 M)) // ler_norm. case=> [r| |]/=. * rewrite /ereal_ball [_ +oo]/= => rM1. apply: MA; rewrite lte_fin. rewrite ger0_norm in rM1; last first. by rewrite subr_ge0 // (le_trans _ (contract_le1 r%:E)) // ler_norm. rewrite ltr_subl_addr addrC addrCA addrC -ltr_subl_addr subrr in rM1. rewrite subr_gt0 in rM1. by rewrite -lte_fin -lt_contract. * by rewrite /ereal_ball /= subrr normr0 => h; exact: MA. * rewrite /ereal_ball /= opprK => h {MA}. exfalso. move: h; apply/negP. rewrite -leNgt [in leRHS]ger0_norm // ler_subl_addr. rewrite -/(contract M%:E) addrC -ler_subl_addr opprD addrA subrr sub0r. by move: (contract_le1 M%:E); rewrite ler_norml => /andP[]. + exists (diag (1 + contract M%:E)%R); rewrite /diag. exists (1 + contract M%:E)%R => //=. rewrite -ltr_subl_addl sub0r. by move: (contract_lt1 M); rewrite ltr_norml => /andP[]. case=> [r| |]. * rewrite /ereal_ball => /= rM1. apply MA. rewrite lte_fin. rewrite ler0_norm in rM1; last first. rewrite ler_subl_addl addr0 ltW //. by move: (contract_lt1 r); rewrite ltr_norml => /andP[]. rewrite opprB opprK -ltr_subl_addl addrK in rM1. by rewrite -lte_fin -lt_contract. * rewrite /ereal_ball /= -opprD normrN => h {MA}. exfalso. move: h; apply/negP. rewrite -leNgt [in leRHS]ger0_norm// -ler_subl_addr addrAC. rewrite subrr add0r -/(contract M%:E). by rewrite (le_trans _ (ltW (contract_lt1 M))) // ler_norm. * rewrite /ereal_ball /= => _; exact: MA. - case: x => [r [E [_/posnumP[e] reA] sEA] | [E [_/posnumP[e] reA] sEA] | [E [_/posnumP[e] reA] sEA]] //=. + by apply nbhs_fin_inbound with e => ? ?; exact/sEA/reA. + have [|] := boolP (e%:num <= 1)%R. by move/nbhs_oo_up_e1; apply => ? ?; exact/sEA/reA. by rewrite -ltNge => /nbhs_oo_up_1e; apply => ? ?; exact/sEA/reA. + have [|] := boolP (e%:num <= 1)%R. by move/nbhs_oo_down_e1; apply => ? ?; exact/sEA/reA. by rewrite -ltNge => /nbhs_oo_down_1e; apply => ? ?; exact/sEA/reA. Qed. Definition ereal_pseudoMetricType_mixin := PseudoMetric.Mixin ereal_ball_center ereal_ball_sym ereal_ball_triangle erefl. Definition ereal_uniformType_mixin : @Uniform.mixin_of (\bar R) nbhs := uniformityOfBallMixin ereal_nbhsE ereal_pseudoMetricType_mixin. Canonical ereal_uniformType := UniformType (extended R) ereal_uniformType_mixin. Canonical ereal_pseudoMetricType := PseudoMetricType (extended R) ereal_pseudoMetricType_mixin. End ereal_PseudoMetric. (* TODO: generalize to numFieldType? *) Lemma lt_ereal_nbhs (R : realFieldType) (a b : \bar R) (r : R) : a < r%:E -> r%:E < b -> exists delta : {posnum R}, forall y, (`|y - r| < delta%:num)%R -> (a < y%:E) && (y%:E < b). Proof. move=> [:wlog]; case: a b => [a||] [b||] //= ltax ltxb. - move: a b ltax ltxb; abstract: wlog. (*BUG*) move=> {}a {}b ltxa ltxb. have m_gt0 : (Num.min ((r - a) / 2) ((b - r) / 2) > 0)%R. by rewrite lt_minr !divr_gt0 // ?subr_gt0. exists (PosNum m_gt0) => y //=; rewrite lt_minr !ltr_distl. move=> /andP[/andP[ay _] /andP[_ yb]]. rewrite 2!lte_fin (lt_trans _ ay) ?(lt_trans yb) //=. rewrite -subr_gt0 opprD addrA {1}[(b - r)%R]splitr addrK. by rewrite divr_gt0 ?subr_gt0. by rewrite -subr_gt0 addrAC {1}[(r - a)%R]splitr addrK divr_gt0 ?subr_gt0. - have [//||d dP] := wlog a (r + 1)%R; rewrite ?lte_fin ?ltr_addl //. by exists d => y /dP /andP[->] /= /lt_le_trans; apply; rewrite leey. - have [//||d dP] := wlog (r - 1)%R b; rewrite ?lte_fin ?gtr_addl ?ltrN10 //. by exists d => y /dP /andP[_ ->] /=; rewrite ltNye. - by exists 1%:pos%R => ? ?; rewrite ltNye ltey. Qed. (* TODO: generalize to numFieldType? *) Lemma nbhs_interval (R : realFieldType) (P : R -> Prop) (r : R) (a b : \bar R) : a < r%:E -> r%:E < b -> (forall y, a < y%:E -> y%:E < b -> P y) -> nbhs r P. Proof. move => ar rb abP; case: (lt_ereal_nbhs ar rb) => d rd. exists d%:num => //= y; rewrite /= distrC. by move=> /rd /andP[? ?]; apply: abP. Qed. Lemma ereal_dnbhs_le (R : numFieldType) (x : \bar R) : ereal_dnbhs x --> ereal_nbhs x. Proof. move: x => [r P [_/posnumP[e] reP] |r P|r P] //=. by exists e%:num => //= ? ? ?; apply: reP. Qed. Lemma ereal_dnbhs_le_finite (R : numFieldType) (r : R) : ereal_dnbhs r%:E --> nbhs r%:E. Proof. by move=> P [_/posnumP[e] reP] //=; exists e%:num => //= ? ? ?; exact: reP. Qed. Definition ereal_loc_seq (R : numDomainType) (x : \bar R) (n : nat) := match x with | x%:E => (x + (n%:R + 1)^-1)%:E | +oo => n%:R%:E | -oo => - n%:R%:E end. Lemma cvg_ereal_loc_seq (R : realType) (x : \bar R) : ereal_loc_seq x --> ereal_dnbhs x. Proof. move=> P; rewrite /ereal_loc_seq. case: x => /= [x [_/posnumP[d] dP] |[d [dreal dP]] |[d [dreal dP]]]; last 2 first. have /ZnatP [N Nfloor] : floor (Num.max d 0%R) \is a Znat. by rewrite Znat_def floor_ge0 le_maxr lexx orbC. exists N.+1 => // n ltNn; apply: dP. have /le_lt_trans : (d <= Num.max d 0)%R by rewrite le_maxr lexx. apply; apply: lt_le_trans (lt_succ_Rfloor _) _; rewrite RfloorE Nfloor. by rewrite -(@natrD R N 1) ler_nat addn1. have /ZnatP [N Nfloor] : floor (Num.max (- d)%R 0%R) \is a Znat. by rewrite Znat_def floor_ge0 le_maxr lexx orbC. exists N.+1 => // n ltNn; apply: dP; rewrite lte_fin ltr_oppl. have /le_lt_trans : (- d <= Num.max (- d) 0)%R by rewrite le_maxr lexx. apply; apply: lt_le_trans (lt_succ_Rfloor _) _; rewrite RfloorE Nfloor. by rewrite -(@natrD R N 1) ler_nat addn1. have /ZnatP [N Nfloor] : floor (d%:num^-1) \is a Znat. by rewrite Znat_def floor_ge0. exists N => // n leNn; have gt0Sn : (0 < n%:R + 1 :> R)%R. by apply: ltr_spaddr => //; exact/ler0n. apply: dP; last first. by rewrite eq_sym addrC -subr_eq subrr eq_sym; apply/invr_neq0/lt0r_neq0. rewrite /= opprD addrA subrr distrC subr0 gtr0_norm; last by rewrite invr_gt0. rewrite -[ltLHS]mulr1 ltr_pdivr_mull // -ltr_pdivr_mulr // div1r. by rewrite (lt_le_trans (lt_succ_Rfloor _))// RfloorE Nfloor ler_add// ler_nat. Qed.