(* 077_sum_of_kth_powers — rocq *)

(*
MIT License

Copyright (c) 2020 Frédéric Chardard, Institut Camille Jordan /
Université Jean Monnet de Saint-Etienne

Permission is hereby granted, free of charge, to any person obtaining a copy
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THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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*)

Require Import Reals.
Open Scope R_scope.


Fixpoint bernoullicoeff (n:nat) (k:nat) : R:=
match n with |  O  => match k with | O => INR 1
                                   | _ => INR 0
                      end
             |(S m)=> 
  match k with 
| O    => (sum_f_R0 (fun l => -((bernoullicoeff m l)*(/INR (S l))*(INR n)*/ INR (S (S l)) )) m)
| (S l)=> (bernoullicoeff m l)*/(INR (S l))*(INR n)
  end
end.


Definition bernoulli_polynomial (n:nat) (x:R):=
(sum_f_R0 (fun l => (bernoullicoeff n l) *x^l) n).


Theorem bernoulli_polynomial_derivable_pt_lim:
forall n:nat,forall x:R,
derivable_pt_lim (bernoulli_polynomial (S n)) x ((bernoulli_polynomial n x)*(INR (S n))).
  intros.
  erewrite (_:(_*(INR (S n))=_)).
 apply derivable_pt_lim_fs.
 apply Nat.lt_0_succ.
 Unshelve.
 unfold Init.Nat.pred.
 rewrite Rmult_comm at 1.
unfold bernoullicoeff.
fold bernoullicoeff.
unfold bernoulli_polynomial.
rewrite scal_sum.
apply sum_eq.
intros.
repeat rewrite<-Rmult_assoc.
field.
apply not_0_INR.
discriminate.
Qed.

Theorem bernoulli_polynomial_derivable_pt_lim_shift:
forall n:nat,forall x:R,
    derivable_pt_lim (fun x=> (bernoulli_polynomial (S n) (x+1))) x ((bernoulli_polynomial n (x+1))*(INR (S n))).
  unfold derivable_pt_lim.
  intros.
destruct (bernoulli_polynomial_derivable_pt_lim n (x+1) eps) as [delta der].
assumption.
exists delta.
intros.
enough(x+h+1=x+1+h) as ->.
apply der;
assumption.
field.
Qed.


Lemma sumzero : forall n:nat, sum_f_R0 (fun _ => 0:R) n=(0:R).
induction n.
simpl sum_f_R0.
reflexivity.
simpl.
rewrite IHn.
apply Rplus_0_r.
Qed.

Theorem bernoulli_polynomial_val_0_1:
  forall n:nat,n<>1%nat  -> bernoulli_polynomial n 0=bernoulli_polynomial n 1.
  intros n nneq1.
  unfold bernoulli_polynomial.
  transitivity ((bernoullicoeff n 0)+0).
destruct n.
unfold sum_f_R0.
field.

rewrite decomp_sum.
unfold Init.Nat.pred.
 rewrite pow_O.
rewrite Rmult_1_r.
f_equal.


rewrite<-(sumzero n) at 1.
apply sum_eq.
intros.
simpl pow.
field.
apply Nat.lt_0_succ.

destruct n.
unfold sum_f_R0,bernoullicoeff.
field.

rewrite decomp_sum.
unfold Init.Nat.pred.
rewrite pow_O.
rewrite Rmult_1_r.
f_equal.
destruct n.
exfalso.
apply nneq1.
reflexivity.
rewrite decomp_sum.
rewrite pow1.
rewrite Rmult_1_r.
simpl Init.Nat.pred.
unfold bernoullicoeff.
fold bernoullicoeff.

unfold bernoullicoeff at 1.
fold bernoullicoeff.
rewrite Rmult_comm.
rewrite<-Rmult_assoc.
rewrite scal_sum.
rewrite Rmult_comm.
rewrite scal_sum.

rewrite<-sum_plus.
rewrite<-(sumzero n) at 1.
apply sum_eq.
intros.
rewrite pow1.
simpl(INR 1).
field.
split;
  apply not_0_INR;
  discriminate.
apply Nat.neq_0_lt_0.
discriminate.
apply Nat.neq_0_lt_0.
discriminate.
Qed.



Lemma primitive_diff : forall f g fp : (R->R), 
(forall x:R, derivable_pt_lim f x (fp x))->
(forall x:R, derivable_pt_lim g x (fp x))->
forall x:R, f x=(plus_fct g (fct_cte ((f 0)-(g 0)))) x.
intros f g fp fder gder.
assert(forall x:R, derivable_pt_lim (f-g) x ((fun _ => 0) x)) as fmpder.
intro x.
rewrite<-(Rminus_diag_eq _ _ (eq_refl (fp x))).
apply derivable_pt_lim_minus.
apply fder.
apply gder.
intro x.
unfold plus_fct,fct_cte.
assert((f-g)%F x=(f-g)%F 0) as diffeq.
destruct (total_order_T x 0) as [xneg|xin];
try destruct xneg as [xin|x0];
try (rewrite x0;
     reflexivity);
pose proof (MVT_cor2 (f-g)%F (fun _ => 0) _ _ xin (fun x _ => fmpder x)) as mvt;
destruct mvt as [c mvt];
destruct mvt as [mvt ineq];
rewrite Rmult_0_l in mvt;
pose proof (Rminus_diag_uniq _ _ mvt) as eqfi;
rewrite eqfi;
reflexivity.
unfold minus_fct in diffeq.
rewrite<-diffeq.
field.
Qed.

Lemma primitive_eq:forall f g fp : (R->R), 
(forall x:R, derivable_pt_lim f x (fp x))->
(forall x:R, derivable_pt_lim g x (fp x))->
f 0=g 0->
forall x:R, f x=g x.
intros f g fp fder gder init x.
transitivity ((plus_fct g (fct_cte ((f 0)-(g 0)))) x).
rewrite (primitive_diff f g fp);
try assumption;
try reflexivity.
unfold plus_fct,fct_cte.
rewrite init.
field.
Qed.

Lemma kthpowers : forall n:nat, forall x:R,
x^n*(INR (S n))=(bernoulli_polynomial (S n) (x+1))-(bernoulli_polynomial (S n) x).
induction n.
intro.
unfold bernoulli_polynomial,sum_f_R0,bernoullicoeff.
unfold sum_f_R0,INR,pow.
field.

apply (primitive_eq _ _ (fun x=>(INR (S n))*x^n*(INR (S (S n)))) ).
intro x.
apply (derivable_pt_lim_scal_right (fun x=>x^(S n)) x _ (INR (S (S n)))).
pose proof (derivable_pt_lim_pow x (S n)) as derpow.
unfold Init.Nat.pred in derpow.
apply derpow.
intro x.
erewrite(_:(INR (S n) * x ^ n * INR (S (S n)))=_).
apply derivable_pt_lim_minus.
apply bernoulli_polynomial_derivable_pt_lim_shift.
apply bernoulli_polynomial_derivable_pt_lim.
unfold pow.
repeat rewrite Rmult_0_l.
rewrite bernoulli_polynomial_val_0_1.
rewrite Rplus_0_l.
rewrite Rminus_diag_eq;
reflexivity.
discriminate.
Unshelve.
rewrite(Rmult_comm (INR (S n))).
rewrite IHn.
field.
Qed.

Lemma sum_kthpowers : forall r:nat, forall n:nat, 
(0<r)%nat->
sum_f_R0 (fun k => ((INR k)^r)%R) n
=((bernoulli_polynomial (S r) (INR n+1))-(bernoulli_polynomial (S r) 0))/(INR r+1).
intros r n rneq0.
induction n.
unfold INR.
fold INR.
rewrite bernoulli_polynomial_val_0_1.
rewrite Rplus_0_l.
unfold sum_f_R0.
unfold pow,INR.
fold INR.
fold pow.
induction r.
exfalso.
apply (Nat.neq_0_lt_0 0).
assumption.
reflexivity.
unfold pow.
fold pow.
field.
rewrite<-S_INR.
apply not_0_INR.
discriminate.
apply not_eq_S.
intro.
apply Nat.neq_0_lt_0 in rneq0.
contradiction.

unfold sum_f_R0.
fold sum_f_R0.
rewrite IHn.
enough(INR (S n) ^ r=(INR (S n) ^r * (INR (S r)))/(INR (S r))) as ->.
rewrite kthpowers.
repeat rewrite<-S_INR.
field.
apply not_0_INR.
discriminate.
field.
apply not_0_INR.
discriminate.
Qed.