Minor clarification of the file organization

Problem: {nngnum _} is in prodnormedzmodule.v where it does not
really belong, the theories of bigmax lemmas using {nngnum _}
does not really belong to normedtype.v

Contents:
- add the `nngnum.v` file for the `{nonneg _}` type and its related
  theories
- minor fixes in `posnum.v`: remove from the documentation the mention
  of `posreal` which seems to have disappeared, typos

NB: prodnormedzmodule.v is a short file that has been
introduced when rebasing on top of MathComp 1.11, there is
another theory of bigmax/bigmin lemmas in normedtype.v that does
not depend on {nngnum _}, let's see latter what to do about these

CHANGELOG_UNRELEASED.md

@@ -62,6 +62,7 @@
   + lemmas `setUK`, `setKU`, `setIK`, `setKI`
   + lemmas `setUK`, `setKU`, `setIK`, `setKI`, `subsetEset`, `subEset`, `complEset`, `botEset`, `topEset`, `meetEset`, `joinEset`, `properEneq`
   + Canonical `porderType`, `latticeType`, `distrLatticeType`, `blatticeType`, `tblatticeType`, `bDistrLatticeType`, `tbDistrLatticeType`, `cbDistrLatticeType`, `ctbDistrLatticeType` on classical `set`.
+- file `nngnum.v`
 
 ### Changed
 
@@ -85,6 +86,7 @@
   + generalization of `lee_sum`
 - in `boolp.v`:
   + rename `exists2NP` to `forall2NP` and make it bidirectionnal
+- moved the definition of `{nngnum _}` and the related bigmax theory to the new `nngnum.v` file
 
 ### Renamed
 

_CoqProject

@@ -5,6 +5,7 @@ theories/boolp.v
 theories/ereal.v
 theories/reals.v
 theories/posnum.v
+theories/nngnum.v
 theories/landau.v
 theories/classical_sets.v
 theories/Rstruct.v

theories/derive.v

@@ -2,7 +2,8 @@
 From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice.
 From mathcomp Require Import ssralg ssrnum fintype bigop order matrix interval.
 Require Import boolp reals.
-Require Import classical_sets posnum topology prodnormedzmodule normedtype landau forms.
+Require Import classical_sets posnum nngnum topology prodnormedzmodule.
+Require Import normedtype landau forms.
 
 (******************************************************************************)
 (* This file provides a theory of differentiation. It includes the standard   *)

theories/landau.v

@@ -3,7 +3,7 @@ From Coq Require Import ssreflect ssrfun ssrbool.
 From mathcomp Require Import ssrnat eqtype choice fintype bigop order ssralg.
 From mathcomp Require Import ssrnum.
 Require Import boolp ereal reals.
-Require Import classical_sets posnum topology normedtype.
+Require Import classical_sets posnum nngnum topology normedtype.
 Require Import prodnormedzmodule.
 
 (******************************************************************************)

theories/nngnum.v

@@ -0,0 +1,318 @@
+From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice.
+From mathcomp Require Import div fintype path bigop order finset fingroup.
+From mathcomp Require Import ssralg poly ssrnum.
+Require Import boolp.
+
+(******************************************************************************)
+(* This file develops tools to make the manipulation of non-negative numbers  *)
+(* easier, on the model of posnum.v. This is WIP.                             *)
+(*                                                                            *)
+(*   {nngnum R} == interface types for elements in R that are non-negative; R *)
+(*                 must have a numDomainType structure. Automatically solves  *)
+(*                 goals of the form x >= 0. {nngnum R} is stable by          *)
+(*                 addition, multiplication. All natural numbers n%:R are     *)
+(*                 also canonically in {nngnum R}.                            *)
+(*                 This type is also shown to be a latticeType, a             *)
+(*                 distrLatticeType, and an orderType,                        *)
+(*  NngNum xge0 == packs the proof xge0 : x >= 0, for x : R, to build a       *)
+(*                 {nngnum R}                                                 *)
+(*       x%:nng == explicitly casts x to {nngnum R}, triggers the inference   *)
+(*                 of a {nngum R} structure for x                             *)
+(*    x%:nngnum == explicit cast from {nngnum R} to R                         *)
+(*                                                                            *)
+(* The module BigmaxrNonneg contains a theory about bigops of the form        *)
+(* \big[maxr/x]_(i | P i) F i where F : I -> {nngnum R} which is used in      *)
+(* normedtype.v to develop the topology of matrices.                          *)
+(*                                                                            *)
+(******************************************************************************)
+
+Set Implicit Arguments.
+Unset Strict Implicit.
+Unset Printing Implicit Defensive.
+
+Local Open Scope ring_scope.
+Import Order.TTheory GRing.Theory Num.Theory.
+
+Reserved Notation "'{nonneg' R }" (at level 0, format "'{nonneg'  R }").
+Reserved Notation "x %:nng" (at level 0, format "x %:nng").
+Reserved Notation "x %:nngnum" (at level 0, format "x %:nngnum").
+Module Nonneg.
+Section nonnegative_numbers.
+
+Record nngnum_of (R : numDomainType) (phR : phant R) := NngNumDef {
+  num_of_nng : R ;
+  nngnum_ge0 :> num_of_nng >= 0
+}.
+Hint Resolve nngnum_ge0 : core.
+Hint Extern 0 ((0 <= _)%R = true) => exact: nngnum_ge0 : core.
+Local Notation "'{nonneg' R }" := (nngnum_of (@Phant R)).
+Definition NngNum (R : numDomainType) x x_ge0 : {nonneg R} :=
+  @NngNumDef _ (Phant R) x x_ge0.
+Arguments NngNum {R}.
+
+Local Notation "x %:nngnum" := (num_of_nng x) : ring_scope.
+Definition nng_of_num (R : numDomainType) (x : {nonneg R})
+   (phx : phantom R x%:nngnum) := x.
+Local Notation "x %:nng" := (nng_of_num (Phantom _ x)) : ring_scope.
+
+Section Order.
+Variable (R : numDomainType).
+
+Canonical nngnum_subType := [subType for @num_of_nng R (Phant R)].
+Definition nngnum_eqMixin := [eqMixin of {nonneg R} by <:].
+Canonical nngnum_eqType := EqType {nonneg R} nngnum_eqMixin.
+Definition nngnum_choiceMixin := [choiceMixin of {nonneg R} by <:].
+Canonical nngnum_choiceType := ChoiceType {nonneg R} nngnum_choiceMixin.
+Definition nngnum_porderMixin := [porderMixin of {nonneg R} by <:].
+Canonical nngnum_porderType :=
+  POrderType ring_display {nonneg R} nngnum_porderMixin.
+
+Lemma nngnum_le_total : totalPOrderMixin [porderType of {nonneg R}].
+Proof. by move=> x y; apply/real_comparable; apply/ger0_real. Qed.
+
+Canonical nngnum_latticeType := LatticeType {nonneg R} nngnum_le_total.
+Canonical nngnum_distrLatticeType := DistrLatticeType {nonneg R} nngnum_le_total.
+Canonical nngnum_orderType := OrderType {nonneg R} nngnum_le_total.
+
+End Order.
+
+End nonnegative_numbers.
+
+Module Exports.
+Arguments NngNum {R}.
+Notation "'{nonneg' R }" := (nngnum_of (@Phant R)) : type_scope.
+Notation nngnum R := (@num_of_nng _ (@Phant R)).
+Notation "x %:nngnum" := (num_of_nng x) : ring_scope.
+Hint Extern 0 ((0 <= _)%R = true) => exact: nngnum_ge0 : core.
+Notation "x %:nng" := (nng_of_num (Phantom _ x)) : ring_scope.
+Canonical nngnum_subType.
+Canonical nngnum_eqType.
+Canonical nngnum_choiceType.
+Canonical nngnum_porderType.
+Canonical nngnum_latticeType.
+Canonical nngnum_orderType.
+End Exports.
+
+End Nonneg.
+Export Nonneg.Exports.
+
+Section NngNum.
+Context {R : numDomainType}.
+Implicit Types a : R.
+Implicit Types x y : {nonneg R}.
+Import Nonneg.
+
+Canonical addr_nngnum x y := NngNum (x%:nngnum + y%:nngnum) (addr_ge0 x y).
+Canonical mulr_nngnum x y := NngNum (x%:nngnum * y%:nngnum) (mulr_ge0 x y).
+Canonical mulrn_nngnum x n := NngNum (x%:nngnum *+ n) (mulrn_wge0 n x).
+Canonical zeror_nngnum := @NngNum R 0 (lexx 0).
+Canonical oner_nngnum := @NngNum R 1 ler01.
+
+Lemma nngnum_lt0 x : (x%:nngnum < 0 :> R) = false.
+Proof. by rewrite le_gtF. Qed.
+
+Lemma nngnum_real x : x%:nngnum \is Num.real.
+Proof. by rewrite ger0_real. Qed.
+Hint Resolve nngnum_real : core.
+
+Lemma nng_eq : {mono nngnum R : x y / x == y}. Proof. by []. Qed.
+Lemma nng_le : {mono nngnum R : x y / x <= y}. Proof. by []. Qed.
+Lemma nng_lt : {mono nngnum R : x y / x < y}. Proof. by []. Qed.
+
+Lemma nng_eq0 x : (x%:nngnum == 0) = (x == 0%:nng).
+Proof. by []. Qed.
+
+Lemma nng_min : {morph nngnum R : x y / Order.min x y}.
+Proof. by move=> x y; rewrite !minEle nng_le -fun_if. Qed.
+
+Lemma nng_max : {morph nngnum R : x y / Order.max x y}.
+Proof. by move=> x y; rewrite !maxEle nng_le -fun_if. Qed.
+
+Lemma nng_le_maxr a x y :
+  a <= Num.max x%:nngnum y%:nngnum = (a <= x%:nngnum) || (a <= y%:nngnum).
+Proof. by rewrite -comparable_le_maxr// real_comparable. Qed.
+
+Lemma nng_le_maxl a x y :
+  Num.max x%:nngnum  y%:nngnum <= a = (x%:nngnum <= a) && (y%:nngnum <= a).
+Proof. by rewrite -comparable_le_maxl// real_comparable. Qed.
+
+Lemma nng_le_minr a x y :
+  a <= Num.min x%:nngnum y%:nngnum = (a <= x%:nngnum) && (a <= y%:nngnum).
+Proof. by rewrite -comparable_le_minr// real_comparable. Qed.
+
+Lemma nng_le_minl a x y :
+  Num.min x%:nngnum y%:nngnum <= a = (x%:nngnum <= a) || (y%:nngnum <= a).
+Proof. by rewrite -comparable_le_minl// real_comparable. Qed.
+
+Lemma max_ge0 x y : Num.max x%:nngnum y%:nngnum >= 0.
+Proof. by rewrite comparable_le_maxr ?real_comparable// ?nngnum_ge0. Qed.
+
+Lemma min_ge0 x y : Num.min x%:nngnum y%:nngnum >= 0.
+Proof. by rewrite comparable_le_minr ?real_comparable// ?nngnum_ge0. Qed.
+
+Canonical max_nngnum x y := NngNum (Num.max x%:nngnum y%:nngnum) (max_ge0 x y).
+Canonical min_nngnum x y := NngNum (Num.min x%:nngnum y%:nngnum) (min_ge0 x y).
+
+Canonical normr_nngnum (V : normedZmodType R) (x : V) :=
+  NngNum `|x| (normr_ge0 x).
+
+Lemma nng_abs_eq0 a : (`|a|%:nng == 0%:nng) = (a == 0).
+Proof. by rewrite -normr_eq0. Qed.
+
+Lemma nng_abs_le a x : 0 <= a -> (`|a|%:nng <= x) = (a <= x%:nngnum).
+Proof. by move=> a0; rewrite -nng_le//= ger0_norm. Qed.
+
+Lemma nng_abs_lt a x : 0 <= a -> (`|a|%:nng < x) = (a < x%:nngnum).
+Proof. by move=> a0; rewrite -nng_lt/= ger0_norm. Qed.
+
+(* Cyril: remove *)
+Lemma nonneg_maxr a x y : `|a| * (Num.max x y)%:nngnum =
+  (Num.max (`|a| * x%:nngnum)%:nng (`|a| * y%:nngnum)%:nng)%:nngnum.
+Proof. by rewrite !nng_max maxr_pmulr. Qed.
+
+End NngNum.
+
+(* TODO: general purpose lemma? add to mathcomp? *)
+Lemma filter_andb I r (a P : pred I) :
+  [seq i <- r | P i && a i] = [seq i <- [seq j <- r | P j] | a i].
+Proof. by elim: r => //= i r ->; case P. Qed.
+
+Import Num.Def.
+
+Module BigmaxrNonneg.
+Section bigmaxr_nonneg.
+Variable (R : numDomainType).
+
+Lemma bigmaxr_mkcond I r (P : pred I) (F : I -> {nonneg R}) x :
+  \big[maxr/x]_(i <- r | P i) F i =
+     \big[maxr/x]_(i <- r) (if P i then F i else x).
+Proof.
+rewrite unlock; elim: r x => //= i r ihr x.
+case P; rewrite ihr // max_r //; elim: r {ihr} => //= j r ihr.
+by rewrite le_maxr ihr orbT.
+Qed.
+
+Lemma bigmaxr_split I r (P : pred I) (F1 F2 : I -> {nonneg R}) x :
+  \big[maxr/x]_(i <- r | P i) (maxr (F1 i) (F2 i)) =
+  maxr (\big[maxr/x]_(i <- r | P i) F1 i) (\big[maxr/x]_(i <- r | P i) F2 i).
+Proof.
+elim/big_rec3: _ => [|i y z _ _ ->]; rewrite ?maxxx //.
+by rewrite maxCA -!maxA maxCA.
+Qed.
+
+Lemma bigmaxr_idl I r (P : pred I) (F : I -> {nonneg R}) x :
+  \big[maxr/x]_(i <- r | P i) F i = maxr x (\big[maxr/x]_(i <- r | P i) F i).
+Proof.
+rewrite -big_filter; elim: [seq i <- r | P i] => [|i l ihl].
+  by rewrite big_nil maxxx.
+by rewrite big_cons maxCA -ihl.
+Qed.
+
+Lemma bigmaxrID I r (a P : pred I) (F : I -> {nonneg R}) x :
+  \big[maxr/x]_(i <- r | P i) F i =
+  maxr (\big[maxr/x]_(i <- r | P i && a i) F i)
+    (\big[maxr/x]_(i <- r | P i && ~~ a i) F i).
+Proof.
+rewrite -!(big_filter _ (fun _ => _ && _)) !filter_andb !big_filter.
+rewrite ![in RHS](bigmaxr_mkcond _ _ F) !big_filter -bigmaxr_split.
+have eqmax : forall i, P i ->
+  maxr (if a i then F i else x) (if ~~ a i then F i else x) = maxr (F i) x.
+  by move=> i _; case: (a i) => //=; rewrite maxC.
+rewrite [RHS](eq_bigr _ eqmax) -!(big_filter _ P).
+elim: [seq j <- r | P j] => [|j l ihl]; first by rewrite !big_nil.
+by rewrite !big_cons -maxA -bigmaxr_idl ihl.
+Qed.
+
+Lemma bigmaxr_seq1 I (i : I) (F : I -> {nonneg R}) x :
+  \big[maxr/x]_(j <- [:: i]) F j = maxr (F i) x.
+Proof. by rewrite big_cons big_nil. Qed.
+
+Lemma bigmaxr_pred1_eq (I : finType) (i : I) (F : I -> {nonneg R}) x :
+  \big[maxr/x]_(j | j == i) F j = maxr (F i) x.
+Proof.
+have [e1 <- _ [e_enum _]] := big_enumP (pred1 i).
+by rewrite (perm_small_eq _ e_enum) enum1 ?bigmaxr_seq1.
+Qed.
+
+Lemma bigmaxr_pred1 (I : finType) i (P : pred I) (F : I -> {nonneg R}) x :
+  P =1 pred1 i -> \big[maxr/x]_(j | P j) F j = maxr (F i) x.
+Proof. by move/(eq_bigl _ _)->; apply: bigmaxr_pred1_eq. Qed.
+
+Lemma bigmaxrD1 (I : finType) j (P : pred I) (F : I -> {nonneg R}) x :
+  P j -> \big[maxr/x]_(i | P i) F i
+    = maxr (F j) (\big[maxr/x]_(i | P i && (i != j)) F i).
+Proof.
+move=> Pj; rewrite (bigmaxrID _ (pred1 j)) [in RHS]bigmaxr_idl maxA.
+by congr maxr; apply: bigmaxr_pred1 => i; rewrite /= andbC; case: eqP => //->.
+Qed.
+
+Lemma ler_bigmaxr_cond (I : finType) (P : pred I) (F : I -> {nonneg R}) x i0 :
+  P i0 -> F i0 <= \big[maxr/x]_(i | P i) F i.
+Proof. by move=> Pi0; rewrite (bigmaxrD1 _ _ Pi0) le_maxr lexx. Qed.
+
+Lemma ler_bigmaxr (I : finType) (F : I -> {nonneg R}) (i0 : I) x :
+  F i0 <= \big[maxr/x]_i F i.
+Proof. exact: ler_bigmaxr_cond. Qed.
+
+Lemma bigmaxr_lerP (I : finType) (P : pred I) m (F : I -> {nonneg R}) x :
+  reflect (x <= m /\ forall i, P i -> F i <= m)
+    (\big[maxr/x]_(i | P i) F i <= m).
+Proof.
+apply: (iffP idP) => [|[lexm leFm]]; last first.
+  by elim/big_ind: _ => // ??; rewrite le_maxl =>->.
+rewrite bigmaxr_idl le_maxl => /andP[-> leFm]; split=> // i Pi.
+by apply: le_trans leFm; apply: ler_bigmaxr_cond.
+Qed.
+
+Lemma bigmaxr_sup (I : finType) i0 (P : pred I) m (F : I -> {nonneg R}) x :
+  P i0 -> m <= F i0 -> m <= \big[maxr/x]_(i | P i) F i.
+Proof. by move=> Pi0 ?; apply: le_trans (ler_bigmaxr_cond _ _ Pi0). Qed.
+
+Lemma bigmaxr_ltrP (I : finType) (P : pred I) m (F : I -> {nonneg R}) x :
+  reflect (x < m /\ forall i, P i -> F i < m)
+    (\big[maxr/x]_(i | P i) F i < m).
+Proof.
+apply: (iffP idP) => [|[ltxm ltFm]]; last first.
+  by elim/big_ind: _ => // ??; rewrite lt_maxl =>->.
+rewrite bigmaxr_idl lt_maxl => /andP[-> ltFm]; split=> // i Pi.
+by apply: le_lt_trans ltFm; apply: ler_bigmaxr_cond.
+Qed.
+
+Lemma bigmaxr_gerP (I : finType) (P : pred I) m (F : I -> {nonneg R}) x :
+  reflect (m <= x \/ exists2 i, P i & m <= F i)
+  (m <= \big[maxr/x]_(i | P i) F i).
+Proof.
+apply: (iffP idP) => [|[lemx|[i Pi lemFi]]]; last 2 first.
+- by rewrite bigmaxr_idl le_maxr lemx.
+- by rewrite (bigmaxrD1 _ _ Pi) le_maxr lemFi.
+rewrite leNgt => /bigmaxr_ltrP /asboolPn.
+rewrite asbool_and negb_and => /orP [/asboolPn/negP|/existsp_asboolPn [i]].
+  by rewrite -leNgt; left.
+by move=> /asboolPn/imply_asboolPn [Pi /negP]; rewrite -leNgt; right; exists i.
+Qed.
+
+Lemma bigmaxr_gtrP (I : finType) (P : pred I) m (F : I -> {nonneg R}) x :
+  reflect (m < x \/ exists2 i, P i & m < F i)
+  (m < \big[maxr/x]_(i | P i) F i).
+Proof.
+apply: (iffP idP) => [|[ltmx|[i Pi ltmFi]]]; last 2 first.
+- by rewrite bigmaxr_idl lt_maxr ltmx.
+- by rewrite (bigmaxrD1 _ _ Pi) lt_maxr ltmFi.
+rewrite ltNge => /bigmaxr_lerP /asboolPn.
+rewrite asbool_and negb_and => /orP [/asboolPn/negP|/existsp_asboolPn [i]].
+  by rewrite -ltNge; left.
+by move=> /asboolPn/imply_asboolPn [Pi /negP]; rewrite -ltNge; right; exists i.
+Qed.
+
+End bigmaxr_nonneg.
+Module Exports.
+Arguments bigmaxr_mkcond {R I r}.
+Arguments bigmaxrID {R I r}.
+Arguments bigmaxr_pred1 {R I} i {P F}.
+Arguments bigmaxrD1 {R I} j {P F}.
+Arguments ler_bigmaxr_cond {R I P F}.
+Arguments ler_bigmaxr {R I F}.
+Arguments bigmaxr_sup {R I} i0 {P m F}.
+End Exports.
+End BigmaxrNonneg.
+Export BigmaxrNonneg.Exports.