Module mathcomp.analysis.numfun

From HB Require Import structures.
From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap.
From mathcomp Require Import mathcomp_extra boolp classical_sets fsbigop.
From mathcomp Require Import functions cardinality set_interval.
Require Import signed reals ereal topology normedtype sequences.


Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import Order.TTheory GRing.Theory Num.Def Num.Theory.
Import numFieldTopology.Exports.

Local Open Scope classical_set_scope.
Local Open Scope ring_scope.

HB.mixin Record isNonNegFun ( aT : Type) ( rT : numDomainType) ( f : aT -> rT) := {
   fun_ge0 : forall x, (0 <= f x)%R
}.
HB.structure Definition NonNegFun aT rT := {f of @isNonNegFun aT rT f}.
Reserved Notation "{ 'nnfun' aT >-> T }"
  (at level 0, format "{ 'nnfun'  aT  >->  T }").
Reserved Notation "[ 'nnfun' 'of' f ]"
  (at level 0, format "[ 'nnfun'  'of'  f ]").
Notation "{ 'nnfun' aT >-> T }" := (@NonNegFun.type aT T) : form_scope.
Notation "[ 'nnfun' 'of' f ]" := [the {nnfun _ >-> _} of f] : form_scope.
#[global] Hint Extern 0 (is_true (0 <= _)) => solve [apply: fun_ge0] : core.

Section fimfun_bin .
Context ( T : Type) ( R : numDomainType).
Variables f g : {fimfun T >-> R}.

Lemma max_fimfun_subproof : @FiniteImage T R (f \max g).
HB.instance Definition _ := max_fimfun_subproof.

End fimfun_bin.

Reserved Notation "f ^\+" (at level 1, format "f ^\+").
Reserved Notation "f ^\-" (at level 1, format "f ^\-").

Section restrict_lemmas .
Context {aT : Type} {rT : numFieldType}.
Implicit Types (f g : aT -> rT) (D : set aT).

Lemma restrict_set0 f : f \_ set0 = cst 0.

Lemma restrict_ge0 D f :
  (forall x, D x -> 0 <= f x) -> forall x, 0 <= (f \_ D) x.

Lemma ler_restrict D f g :
  (forall x, D x -> f x <= g x) -> forall x, (f \_ D) x <= (g \_ D) x.

End restrict_lemmas.

Lemma erestrict_ge0 {aT} {rT : numFieldType} ( D : set aT) ( f : aT -> \bar rT) :
  (forall x, D x -> (0 <= f x)%E) -> forall x, (0 <= (f \_ D) x)%E.

Lemma lee_restrict {aT} {rT : numFieldType} ( D : set aT) ( f g : aT -> \bar rT) :
  (forall x, D x -> f x <= g x)%E -> forall x, ((f \_ D) x <= (g \_ D) x)%E.

Lemma restrict_lee {aT} {rT : numFieldType} ( D E : set aT) ( f : aT -> \bar rT) :
  (forall x, E x -> 0 <= f x)%E ->
  D `<=` E -> forall x, ((f \_ D) x <= (f \_ E) x)%E.

Section erestrict_lemmas .
Local Open Scope ereal_scope.
Variables ( T : Type) ( R : realDomainType) ( D : set T).
Implicit Types (f g : T -> \bar R) (r : R).

Lemma erestrict_set0 f : f \_ set0 = cst 0.

Lemma erestrict0 : (cst 0 : T -> \bar R) \_ D = cst 0.

Lemma erestrictD f g : (f \+ g) \_ D = f \_ D \+ g \_ D.

Lemma erestrictN f : (\- f) \_ D = \- f \_ D.

Lemma erestrictB f g : (f \- g) \_ D = f \_ D \- g \_ D.

Lemma erestrictM f g : (f \* g) \_ D = f \_ D \* g \_ D.

Lemma erestrict_scale k f :
  (fun x => k%:E * f x) \_ D = (fun x => k%:E * (f \_ D) x).

End erestrict_lemmas.

Section funposneg .
Local Open Scope ereal_scope.

Definition funepos T ( R : realDomainType) ( f : T -> \bar R) :=
  fun x => maxe (f x) 0.
Definition funeneg T ( R : realDomainType) ( f : T -> \bar R) :=
  fun x => maxe (- f x) 0.

End funposneg.

Notation "f ^\+" := (funepos f) : ereal_scope.
Notation "f ^\-" := (funeneg f) : ereal_scope.

Section funposneg_lemmas .
Local Open Scope ereal_scope.
Variables ( T : Type) ( R : realDomainType) ( D : set T).
Implicit Types (f g : T -> \bar R) (r : R).

Lemma funepos_ge0 f x : 0 <= f^\+ x.

Lemma funeneg_ge0 f x : 0 <= f^\- x.

Lemma funeposN f : (\- f)^\+ = f^\-

Lemma funenegN f : (\- f)^\- = f^\+.

Lemma funepos_restrict f : (f \_ D)^\+ = (f^\+) \_ D.

Lemma funeneg_restrict f : (f \_ D)^\- = (f^\-) \_ D.

Lemma ge0_funeposE f : (forall x, D x -> 0 <= f x) -> {in D, f^\+ =1 f}.

Lemma ge0_funenegE f : (forall x, D x -> 0 <= f x) -> {in D, f^\- =1 cst 0}.

Lemma le0_funeposE f : (forall x, D x -> f x <= 0) -> {in D, f^\+ =1 cst 0}.

Lemma le0_funenegE f : (forall x, D x -> f x <= 0) -> {in D, f^\- =1 \- f}.

Lemma gt0_funeposM r f : (0 < r)%R ->
  (fun x => r%:E * f x)^\+ = (fun x => r%:E * (f^\+ x)).

Lemma gt0_funenegM r f : (0 < r)%R ->
  (fun x => r%:E * f x)^\- = (fun x => r%:E * (f^\- x)).

Lemma lt0_funeposM r f : (r < 0)%R ->
  (fun x => r%:E * f x)^\+ = (fun x => - r%:E * (f^\- x)).

Lemma lt0_funenegM r f : (r < 0)%R ->
  (fun x => r%:E * f x)^\- = (fun x => - r%:E * (f^\+ x)).

Lemma fune_abse f : abse \o f = f^\+ \+ f^\-.

Lemma funeposneg f : f = (fun x => f^\+ x - f^\- x).

Lemma add_def_funeposneg f x : (f^\+ x +? - f^\- x).

Lemma funeD_Dpos f g : f \+ g = (f \+ g)^\+ \- (f \+ g)^\-.

Lemma funeD_posD f g : f \+ g = (f^\+ \+ g^\+) \- (f^\- \+ g^\-).

End funposneg_lemmas.
#[global]
Hint Extern 0 (is_true (0 <= _ ^\+ _)%E) => solve [apply: funepos_ge0] : core.
#[global]
Hint Extern 0 (is_true (0 <= _ ^\- _)%E) => solve [apply: funeneg_ge0] : core.

Definition indic {T} {R : ringType} ( A : set T) ( x : T) : R := (x \in A)%:R.
Reserved Notation "'\1_' A" (at level 8, A at level 2, format "'\1_' A") .
Notation "'\1_' A" := (indic A) : ring_scope.

Section indic_lemmas .
Context ( T : Type) ( R : ringType).
Implicit Types A D : set T.

Lemma indicE A ( x : T) : \1_A x = (x \in A)%:R :> R

Lemma indicT : \1_[set: T] = cst (1 : R).

Lemma indic0 : \1_(@set0 T) = cst (0 : R).

Lemma preimage_indic D ( B : set R) :
  \1_D @^-1` B = if 1 \in B then (if 0 \in B then setT else D)
                            else (if 0 \in B then ~` D else set0).

Lemma image_indic D A :
  \1_D @` A = (if A `\` D != set0 then [set 0] else set0) `|`
              (if A `&` D != set0 then [set 1 : R] else set0).

Lemma image_indic_sub D A : \1_D @` A `<=` ([set 0; 1] : set R).

Lemma fimfunE ( f : {fimfun T >-> R}) x :
  f x = \sum_( y \in range f) (y * \1_(f @^-1` [set y]) x).

End indic_lemmas.

Lemma patch_indic T {R : numFieldType} ( f : T -> R) ( D : set T) :
  f \_ D = (f \* \1_D)%R.

Lemma xsection_indic ( R : ringType) T1 T2 ( A : set (T1 * T2)) x :
  xsection A x = (fun y => (\1_A (x, y) : R)) @^-1` [set 1].

Lemma ysection_indic ( R : ringType) T1 T2 ( A : set (T1 * T2)) y :
  ysection A y = (fun x => (\1_A (x, y) : R)) @^-1` [set 1].

Lemma indic_restrict {T : pointedType} {R : numFieldType} ( A : set T) :
  \1_A = 1 \_ A :> (T -> R).

Lemma restrict_indic T ( R : numFieldType) ( E A : set T) :
  (\1_E \_ A) = \1_(E `&` A) :> (T -> R).

Section ring .
Context ( aT : pointedType) ( rT : ringType).

Lemma fimfun_mulr_closed : mulr_closed (@fimfun aT rT).
Canonical fimfun_mul := MulrPred fimfun_mulr_closed.
Canonical fimfun_ring := SubringPred fimfun_mulr_closed.
Definition fimfun_ringMixin := [ringMixin of {fimfun aT >-> rT} by <:].
Canonical fimfun_ringType := RingType {fimfun aT >-> rT} fimfun_ringMixin.

Implicit Types (f g : {fimfun aT >-> rT}).

Lemma fimfunM f g : f * g = f \* g :> (_ -> _)
Lemma fimfun1 : (1 : {fimfun aT >-> rT}) = cst 1 :> (_ -> _)
Lemma fimfun_prod I r ( P : {pred I}) ( f : I -> {fimfun aT >-> rT}) ( x : aT) :
  (\sum_( i <- r | P i) f i) x = \sum_( i <- r | P i) f i x.
Lemma fimfunX f n : f ^+ n = (fun x => f x ^+ n) :> (_ -> _).

Lemma indic_fimfun_subproof X : @FiniteImage aT rT \1_X.
HB.instance Definition _ X := indic_fimfun_subproof X.
Definition indic_fimfun ( X : set aT) := [the {fimfun aT >-> rT} of \1_X].

HB.instance Definition _ k f := FImFun.copy (k \o* f) (f * cst_fimfun k).
Definition scale_fimfun k f := [the {fimfun aT >-> rT} of k \o* f].

End ring.
Arguments indic_fimfun {aT rT} _.

Section comring .
Context ( aT : pointedType) ( rT : comRingType).
Definition fimfun_comRingMixin := [comRingMixin of {fimfun aT >-> rT} by <:].
Canonical fimfun_comRingType :=
  ComRingType {fimfun aT >-> rT} fimfun_comRingMixin.

Implicit Types (f g : {fimfun aT >-> rT}).
HB.instance Definition _ f g := FImFun.copy (f \* g) (f * g).
End comring.

HB.factory Record FiniteDecomp ( T : pointedType) ( R : ringType) ( f : T -> R) :=
  { fimfunE : exists ( r : seq R) ( A_ : R -> set T),
      forall x, f x = \sum_( y <- r) (y * \1_(A_ y) x) }.
HB.builders Context T R f of @FiniteDecomp T R f.
  Lemma finite_subproof: @FiniteImage T R f.
  HB.instance Definition _ := finite_subproof.
HB.end.

Section Tietze .
Context {X : topologicalType} {R : realType}.

Local Notation "3" := 3%:R : ring_scope.

Hypothesis normalX : normal_space X.

Lemma urysohn_ext_itv A B x y :
  closed A -> closed B -> A `&` B = set0 -> x < y ->
  exists f : X -> R, [/\ continuous f,
    f @` A `<=` [set x], f @` B `<=` [set y] & range f `<=` `[x, y]].

Context ( A : set X).
Hypothesis clA : closed A.

Local Notation "3" := 3%:R.

Local Lemma tietze_step' ( f : X -> R) ( M : R) :
  0 < M -> {within A, continuous f} ->
  (forall x, A x -> `|f x| <= M) ->
  exists g : X -> R, [/\ continuous g,
     (forall x, A x -> `|f x - g x| <= 2/3 * M) &
     (forall x, `|g x| <= 1/3 * M)].

Let tietze_step ( f : X -> R) M :
  {g : X -> R^o | {within A, continuous f} -> 0 < M ->
    (forall x, A x -> `|f x| <= M) -> [/\ continuous g,
      forall x, A x -> `|f x - g x| <= 2/3 * M :>R
      & forall x, `|g x| <= 1/3 * M ]}.

Let onem_twothirds : 1 - 2/3%:R = 1/3%:R :> R.

Lemma continuous_bounded_extension ( f : X -> R^o) M :
  0 < M -> {within A, continuous f} -> (forall x, A x -> `|f x| <= M) ->
  exists g, [/\ {in A, f =1 g}, continuous g & forall x, `|g x| <= M].

End Tietze.