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Polynomial.v
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Require Import Psatz.
Require Import String.
Require Export Complex.
Require Import List.
Require Import Setoid.
Declare Scope poly_scope.
Delimit Scope poly_scope with P.
Open Scope poly_scope.
(* some prelim lemmas that should be moved *)
Lemma diff_pow_n : forall (n : nat) (a b : C),
(a^(S n) - b^(S n)) = (a - b) * (big_sum (fun i => a^i * b^(n - i)) (S n)).
Proof. intros.
unfold Cminus.
rewrite Cmult_plus_distr_r.
do 2 rewrite (@big_sum_mult_l C _ _ _ C_is_ring).
rewrite <- big_sum_extend_r, <- big_sum_extend_l, Nat.sub_diag, Nat.sub_0_r.
simpl.
replace (a * (a ^ n * C1)) with (a^ (S n)) by lca.
replace (- b * (C1 * b ^ n)) with (- b^(S n)) by lca.
rewrite (Cplus_comm _ (a ^ _)), <- Cplus_assoc.
apply Cplus_simplify; try easy.
rewrite Cplus_assoc, (Cplus_comm _ (- b ^ _)), <- (Cplus_0_r (- b ^ _)).
do 2 rewrite <- Cplus_assoc.
rewrite <- (Cplus_0_r (- (b * b ^ n))).
apply Cplus_simplify; try easy.
rewrite <- (@big_sum_plus C _ _ C_is_comm_group), Cplus_0_l.
rewrite big_sum_0_bounded; auto.
intros.
replace (n - x)%nat with (S (n - S x))%nat by lia.
lca.
Qed.
Lemma destruct_from_last : forall {X} (n : nat) (d : X) (l : list X),
length l = S n -> l = (firstn n l) ++ [nth n l d].
Proof. induction n as [| n'].
- intros.
repeat (destruct l; try easy).
- intros.
destruct l; try easy.
simpl in *.
apply Nat.succ_inj in H.
rewrite <- IHn'; easy.
Qed.
Lemma ind_from_end_helper : forall {X} (p : list X -> Prop) (n : nat),
p [] ->
(forall (x : X) (l : list X), p l -> p (l ++ [x])) ->
forall l, length l = n -> p l.
Proof. intros X p n H H0.
induction n as [| n'].
- intros.
destruct l; try easy.
- intros. destruct l; try easy.
rewrite (destruct_from_last n' x); auto.
apply H0.
apply IHn'.
rewrite firstn_length, H1.
rewrite min_l; lia.
Qed.
Lemma ind_from_end : forall {X} (p : list X -> Prop),
p [] ->
(forall (x : X) (l : list X), p l -> p (l ++ [x])) ->
forall l, p l.
Proof. intros.
apply (ind_from_end_helper _ (length l)); easy.
Qed.
(* polynomial represented by a list of coeficients and a degree*)
Definition Polynomial := list C.
Definition Peval (p : Polynomial) (x : Complex.C):=
big_sum (fun i => (nth i p C0)* x^i) (length p).
Definition Peq (p1 p2 : Polynomial) :=
Peval p1 = Peval p2.
Fixpoint Pplus (p1 p2 : Polynomial) : Polynomial :=
match p1 with
| [] => p2
| (a1 :: p1') =>
match p2 with
| [] => p1
| (a2 :: p2') => (a1 + a2) :: (Pplus p1' p2')
end
end.
Fixpoint Pmult (p1 p2 : Polynomial) : Polynomial :=
match p1 with
| [] => []
| (a1 :: p1') => Pplus (map (Cmult a1) p2) (C0 :: (Pmult p1' p2))
end.
Definition Popp (p : Polynomial) : Polynomial :=
Pmult [-C1] p.
Global Program Instance P_is_monoid : Monoid Polynomial :=
{ Gzero := []
; Gplus := Pplus
}.
Solve All Obligations with program_simpl; try lca.
Next Obligation. induction g as [| g']; easy. Qed.
Next Obligation.
assert (H' : forall p1 p2 p3, Pplus (Pplus p1 p2) p3 = Pplus p1 (Pplus p2 p3)).
induction p1 as [| a]; try easy.
intros.
destruct p2; destruct p3; try easy; simpl.
rewrite IHp1, Cplus_assoc; easy.
rewrite H'; easy.
Qed.
(*
Fixpoint Psum (f : nat -> Polynomial) (n : nat) : Polynomial :=
match n with
| 0 => [C0]
| S n' => (Pplus (Psum f n') (f n'))
end.
*)
Infix "≅" := Peq (at level 70) : poly_scope.
Infix "+," := Pplus (at level 50, left associativity) : poly_scope.
Infix "*," := Pmult (at level 40, left associativity) : poly_scope.
Notation "-, P" := (Popp P) (at level 35) : poly_scope.
Notation "P [[ x ]]" := (Peval P x) (at level 0) : poly_scope.
(* some useful lemmas about Peval *)
Lemma cons_eval : forall (p1 : Polynomial) (a c : C),
(a :: p1)[[c]] = a + c * (p1[[c]]).
Proof. intros.
unfold Peval.
simpl length.
rewrite <- big_sum_extend_l.
repeat apply f_equal_gen; try easy; try lca.
rewrite (@big_sum_mult_l C _ _ _ C_is_ring); apply big_sum_eq_bounded.
intros.
lca.
Qed.
Lemma Pplus_eval : forall (p1 p2 : Polynomial) (c : C),
(p1 +, p2)[[c]] = p1[[c]] + p2[[c]].
Proof. induction p1 as [| a1].
- intros.
unfold Peval; lca.
- intros.
destruct p2.
+ unfold Peval; lca.
+ simpl in *.
do 3 rewrite cons_eval.
rewrite IHp1; lca.
Qed.
Lemma Pmult_eval : forall (p1 p2 : Polynomial) (c : C),
(p1 *, p2)[[c]] = p1[[c]] * p2[[c]].
Proof. induction p1 as [| a1].
- intros; lca.
- intros; simpl.
rewrite Pplus_eval.
do 2 rewrite cons_eval.
rewrite IHp1, Cmult_plus_distr_r.
apply f_equal_gen; try lca.
apply f_equal_gen; try easy.
unfold Peval.
rewrite map_length, (@big_sum_mult_l C _ _ _ C_is_ring).
apply big_sum_eq_bounded; intros.
replace C0 with (Cmult a1 C0) by lca.
rewrite map_nth, Cmult_0_r.
lca.
Qed.
Lemma Psum_eval : forall (f : nat -> Polynomial) (n : nat) (c : C),
(big_sum f n)[[c]] = big_sum (fun i => (f i)[[c]]) n.
Proof. induction n as [| n'].
- intros. unfold Peval; lca.
- intros; simpl.
rewrite Pplus_eval.
repeat apply f_equal_gen; try easy; try lca.
Qed.
Lemma app_eval : forall (f g : Polynomial),
Peval (f ++ g) = (fun c => f [[ c ]] + ((repeat C0 (length f)) ++ g) [[ c ]]).
Proof. intros.
apply functional_extensionality; intros.
unfold Peval.
rewrite app_length, big_sum_sum.
apply Cplus_simplify.
apply big_sum_eq_bounded; intros.
rewrite app_nth1; easy.
rewrite app_length, big_sum_sum.
rewrite repeat_length.
assert (big_sum (fun i : nat => nth i (repeat C0 (Datatypes.length f) ++ g) C0 * x ^ i)
(length f) = C0).
{ apply (@big_sum_0_bounded C C_is_monoid); intros.
rewrite app_nth1.
destruct (nth_in_or_default x0 (repeat C0 (Datatypes.length f)) C0).
- apply repeat_spec in i.
rewrite i; lca.
- rewrite e; lca.
- rewrite repeat_length; easy. }
rewrite H, Cplus_0_l.
apply big_sum_eq_bounded; intros.
rewrite app_nth2_plus, app_nth2, repeat_length.
replace (Datatypes.length f + x0 - Datatypes.length f)%nat with x0 by lia.
easy.
rewrite repeat_length; lia.
Qed.
Lemma nth_repeat : forall {X} (x : X) (i n : nat),
(nth i (repeat x n) x) = x.
Proof. induction i as [| i'].
- destruct n; easy.
- destruct n; try easy.
simpl in *.
apply IHi'.
Qed.
Lemma mul_by_x_to_n : forall (f : Polynomial) (n : nat) (c : C),
((repeat C0 n) ++ f)[[c]] = f[[c]] * c^n.
Proof. intros.
unfold Peval.
rewrite app_length, big_sum_sum, <- Cplus_0_l.
apply Cplus_simplify.
apply (@big_sum_0_bounded C C_is_monoid); intros.
rewrite app_nth1, nth_repeat; auto; lca.
rewrite (@big_sum_mult_r C _ _ _ C_is_ring).
apply big_sum_eq_bounded; intros.
rewrite app_nth2_plus, repeat_length, Cpow_add_r; lca.
Qed.
Lemma app_eval_to_mul : forall (f g : Polynomial) (c : C),
(f ++ g)[[c]] = f[[c]] + c^(length f) * g[[c]].
Proof. intros.
rewrite Cmult_comm, app_eval, mul_by_x_to_n; easy.
Qed.
(* Now we show that Peq is an equivalence relation, and we prove compatability lemmas *)
Lemma Peq_refl : forall (p : Polynomial), p ≅ p.
Proof. easy. Qed.
Lemma Peq_symm : forall (p1 p2 : Polynomial), p1 ≅ p2 -> p2 ≅ p1.
Proof. easy. Qed.
Lemma Peq_trans : forall (p1 p2 p3 : Polynomial),
p1 ≅ p2 -> p2 ≅ p3 -> p1 ≅ p3.
Proof. intros.
unfold Peq in *; rewrite H; easy.
Qed.
Lemma Pplus_eq_compat : forall (p1 p1' p2 p2' : Polynomial),
p1 ≅ p1' -> p2 ≅ p2' -> p1 +, p2 ≅ p1' +, p2'.
Proof. intros.
unfold Peq in *.
apply functional_extensionality; intros.
do 2 rewrite Pplus_eval.
rewrite H, H0; easy.
Qed.
Lemma Pmult_eq_compat : forall (p1 p1' p2 p2' : Polynomial),
p1 ≅ p1' -> p2 ≅ p2' -> p1 *, p2 ≅ p1' *, p2'.
Proof. intros.
unfold Peq in *.
apply functional_extensionality; intros.
do 2 rewrite Pmult_eval.
rewrite H, H0; easy.
Qed.
Add Parametric Relation : Polynomial Peq
reflexivity proved by Peq_refl
symmetry proved by Peq_symm
transitivity proved by Peq_trans
as Peq_equiv_rel.
Add Parametric Morphism : Pplus
with signature Peq ==> Peq ==> Peq as Pplus_mor.
Proof. intros p1 p2 H p1' p2' H0; apply Pplus_eq_compat; easy. Qed.
Add Parametric Morphism : Pmult
with signature Peq ==> Peq ==> Peq as Pmult_mor.
Proof. intros p1 p2 H p1' p2' H0; apply Pmult_eq_compat; easy. Qed.
Add Parametric Morphism : cons
with signature eq ==> Peq ==> Peq as cons_mor.
Proof. intros c p1 p2 H.
unfold Peq in *.
apply functional_extensionality; intros.
do 2 rewrite cons_eval.
rewrite H; easy.
Qed.
(* now we prove some basic lemmas *)
Lemma Pplus_comm : forall (p1 p2 : Polynomial),
p1 +, p2 = p2 +, p1.
Proof. induction p1 as [| h];
intros; destruct p2; try easy; simpl.
rewrite IHp1, Cplus_comm; easy.
Qed.
Lemma Pplus_assoc : forall (p1 p2 p3 : Polynomial),
(p1 +, p2) +, p3 = p1 +, (p2 +, p3).
Proof. induction p1 as [| h]; try easy.
intros.
destruct p2; destruct p3; try easy; simpl.
rewrite IHp1, Cplus_assoc; easy.
Qed.
Lemma Pplus_0_r : forall (p : Polynomial),
p +, [] = p.
Proof. intros.
destruct p; easy.
Qed.
Lemma C0_Peq_nil : [C0] ≅ [].
Proof. apply functional_extensionality; intros; lca. Qed.
Lemma Pmult_0_r : forall (p : Polynomial),
p *, [] ≅ [].
Proof. induction p as [| a]; try easy.
simpl.
rewrite IHp.
apply C0_Peq_nil.
Qed.
Lemma Pscale_comm_nempty : forall (a : C) (p : Polynomial),
p <> [] -> [a] *, p = p *, [a].
Proof. induction p as [| h]; try easy.
intros; simpl.
destruct p.
+ simpl; rewrite Cmult_comm; easy.
+ rewrite <- IHp; try easy.
apply f_equal_gen; try (apply f_equal_gen; try lca; easy).
simpl.
rewrite Pplus_0_r, Cplus_0_r.
reflexivity.
Qed.
Lemma Pmult_comm_helper : forall (n : nat) (p1 p2 : Polynomial),
(length p1 + length p2 <= n)%nat ->
p1 *, p2 ≅ p2 *, p1.
Proof. induction n as [| n'].
- intros.
destruct p1; destruct p2; try easy.
- intros.
destruct p1; destruct p2; try (rewrite Pmult_0_r; easy).
simpl in *.
rewrite (IHn' p1 (c0 :: p2)), (IHn' p2 (c :: p1)); simpl; try lia.
rewrite (IHn' p1 p2); try lia.
do 2 rewrite <- Pplus_assoc.
rewrite (Pplus_comm _ (map (Cmult c0) p1)), Cmult_comm; easy.
Qed.
Lemma Pmult_comm : forall (p1 p2 : Polynomial),
p1 *, p2 ≅ p2 *, p1.
Proof. intros.
apply (Pmult_comm_helper (length p1 + length p2)).
easy.
Qed.
Lemma map_comm_distr_l : forall (p1 p2 : Polynomial) (c : C),
map (Cmult c) (p1 +, p2) = map (Cmult c) p1 +, map (Cmult c) p2.
Proof. induction p1 as [| a1]; try easy.
intros.
destruct p2; try easy; simpl.
rewrite IHp1.
repeat (apply f_equal_gen; try lca); easy.
Qed.
Lemma Pmult_plus_distr_l : forall (p1 p2 p3 : Polynomial),
p1 *, (p2 +, p3) ≅ p1 *, p2 +, p1 *, p3.
Proof. induction p1 as [| a1]; try easy.
intros.
simpl. rewrite IHp1.
rewrite map_comm_distr_l.
unfold Peq; apply functional_extensionality; intros.
repeat (rewrite Pplus_eval).
repeat (rewrite <- Cplus_assoc).
apply f_equal_gen; try easy.
repeat rewrite cons_eval.
rewrite Pplus_eval.
lca.
Qed.
Lemma Pmult_plus_distr_r : forall (p1 p2 p3 : Polynomial),
(p2 +, p3) *, p1 ≅ p2 *, p1 +, p3 *, p1.
Proof. intros.
rewrite Pmult_comm, Pmult_plus_distr_l.
rewrite Pmult_comm, (Pmult_comm _ p3).
easy.
Qed.
Lemma cons_simplify : forall (p : Polynomial) (a : C),
(a :: p) = [a] +, (C0 :: p).
Proof. intros.
unfold Pplus.
rewrite Cplus_0_r; easy.
Qed.
Lemma cons_0_mult : forall (p1 p2 : Polynomial),
(C0 :: p1) *, p2 ≅ C0 :: (p1 *, p2).
Proof. intros; simpl.
assert (H' : (map (Cmult C0) p2) ≅ ([C0] *, p2)).
{ simpl. rewrite C0_Peq_nil, Pplus_comm; easy. }
rewrite H', C0_Peq_nil.
easy.
Qed.
Lemma cons_singleton_mult : forall (p : Polynomial) (a : C),
[a] *, p ≅ map (Cmult a) p.
Proof. intros; simpl.
rewrite C0_Peq_nil, Pplus_comm; easy.
Qed.
Lemma Pmult_assoc_singleton : forall (a1 : C) (p2 p3 : Polynomial),
([a1] *, p2) *, p3 ≅ [a1] *, (p2 *, p3).
Proof. induction p2 as [| a2].
- intros.
rewrite Pmult_0_r; simpl.
rewrite C0_Peq_nil; easy.
- intros.
rewrite (cons_simplify p2 a2), Pmult_plus_distr_l,
Pmult_plus_distr_r, (Pmult_comm _ (C0 :: p2)).
do 2 rewrite cons_0_mult.
rewrite (Pmult_comm p2), IHp2, Pmult_plus_distr_r, Pmult_plus_distr_l, cons_0_mult.
rewrite (Pmult_comm _ (C0 :: p2 *, p3)), cons_0_mult,
(Pmult_comm _ [a1]), cons_singleton_mult; simpl map.
rewrite cons_singleton_mult, (cons_singleton_mult _ a2),
(cons_singleton_mult (map (Cmult a2) p3)), map_map.
repeat (apply f_equal_gen; try easy).
apply functional_extensionality; intros.
lca.
Qed.
Lemma Pmult_assoc : forall (p1 p2 p3 : Polynomial),
(p1 *, p2) *, p3 ≅ p1 *, (p2 *, p3).
Proof. induction p1 as [| a1]; try easy.
intros.
rewrite cons_simplify.
do 3 rewrite Pmult_plus_distr_r, cons_0_mult.
rewrite Pmult_assoc_singleton, IHp1.
easy.
Qed.
Lemma map_1_id : forall (p : Polynomial),
map (Cmult C1) p = p.
Proof. induction p; try easy.
simpl.
rewrite IHp, Cmult_1_l.
easy.
Qed.
Lemma Pmult_1_l : forall (p : Polynomial),
[C1] *, p ≅ p.
Proof. intros.
simpl.
rewrite map_1_id, C0_Peq_nil, Pplus_0_r.
easy.
Qed.
Lemma Pmult_1_r : forall (p : Polynomial),
p *, [C1] ≅ p.
Proof. intros.
rewrite Pmult_comm.
apply Pmult_1_l.
Qed.
Lemma Pplus_opp_r : forall (p : Polynomial),
p +, -,p ≅ [].
Proof. induction p as [| a].
- unfold Popp; simpl.
rewrite C0_Peq_nil; easy.
- simpl.
assert (H' : (map (Cmult (- C1)) p +, []) ≅ -, p).
{ unfold Popp, Pmult;
rewrite C0_Peq_nil; easy. }
rewrite H', IHp.
replace (a + (- C1 * a + C0)) with C0 by lca.
rewrite C0_Peq_nil; easy.
Qed.
Lemma Pplus_opp_l : forall (p : Polynomial),
-,p +, p ≅ [].
Proof. intros.
rewrite Pplus_comm.
apply Pplus_opp_r.
Qed.
Lemma head_0_Pmult_x : forall (p : Polynomial),
(C0 :: p) ≅ [C0; C1] *, p.
Proof. intros.
rewrite cons_0_mult, Pmult_1_l.
easy.
Qed.
(* We now need to show that if two polynomials are equal at infinite values,
then they are infinite everywhere *)
(* we start by defining limits *)
Definition limit_at_point (f : C -> C) (a L : C) : Prop :=
forall ϵ : R, ϵ > 0 -> exists δ, (δ > 0 /\ forall x, x <> a -> Cmod (x - a) < δ -> Cmod (f(x) - L) < ϵ).
Definition continuous_at (f : C -> C) (a : C) : Prop :=
limit_at_point f a (f a).
Lemma limit_unique : forall (f : C -> C) (a L1 L2 : C),
limit_at_point f a L1 -> limit_at_point f a L2 ->
L1 = L2.
Proof. intros.
destruct (Ceq_dec (L1 - L2) C0).
- rewrite <- Cplus_0_r, <- e.
lca.
- unfold limit_at_point in *.
destruct (H ((Cmod (L1 - L2)) / 2)%R) as [δ1 [H1 H2] ];
destruct (H0 ((Cmod (L1 - L2)) / 2)%R) as [δ2 [H3 H4] ];
try (unfold Rdiv; apply Rmult_gt_0_compat; try lra; apply Cmod_gt_0; easy).
assert (H5 : exists b, b <> a /\ Cmod (b - a) < (Rmin δ1 δ2)).
{ exists (a + ((Rmin δ1 δ2) / 2, 0)%R)%C.
split.
+ unfold not; intros.
apply (f_equal (Cplus (-a))) in H5.
rewrite Cplus_assoc, Cplus_opp_l, Cplus_0_l in H5.
apply (f_equal fst) in H5; simpl in H5.
apply (f_equal (Rmult 2)) in H5.
rewrite Rmult_0_r in H5.
replace (2 * (Rmin δ1 δ2 / 2))%R with (Rmin δ1 δ2) in H5 by lra.
apply (Rmin_pos δ1 δ2) in H1; auto.
lra.
+ unfold Cplus, Cminus, Copp, Cmod; simpl.
replace (snd a + 0 + - snd a)%R with 0 by lra.
replace (fst a + Rmin δ1 δ2 / 2 + - fst a)%R with (Rmin δ1 δ2 / 2)%R by lra.
rewrite Rmult_0_l, Rplus_0_r, Rmult_1_r, sqrt_square.
apply Rlt_eps2_eps.
apply (Rmin_pos δ1 δ2); auto.
unfold Rdiv.
apply Rmult_le_pos; try lra.
left; apply (Rmin_pos δ1 δ2); auto. }
destruct H5 as [b [H5 H6] ].
assert (H7 : Cmod (f b - L1) + Cmod (f b - L2) < Cmod (L1 - L2)).
rewrite (double_var (Cmod (L1 - L2))).
apply Rplus_lt_compat.
apply H2; auto.
assert (H' := (Rmin_l δ1 δ2)). lra.
apply H4; auto.
assert (H' := (Rmin_r δ1 δ2)). lra.
assert (H8 : Cmod (L1 - (f b) + (f b - L2)) <= Cmod (L1 - f b) + Cmod (f b - L2)).
{ apply Cmod_triangle. }
replace (L1 - f b + (f b - L2)) with (L1 - L2) in H8 by lca.
replace (L1 - f b) with (- (f b - L1)) in H8 by lca.
rewrite Cmod_opp in H8. lra.
Qed.
Lemma limit_const : forall (a x : C),
limit_at_point (fun c => a) x a.
Proof. unfold limit_at_point; intros.
exists 1. split; try lra.
intros.
replace (a - a) with C0 by lca.
rewrite Cmod_0.
easy.
Qed.
Lemma limit_const_poly : forall (a x : C),
limit_at_point (Peval [a]) x a.
Proof. unfold limit_at_point.
intros.
exists 1; split.
lra.
intros.
unfold Peval; simpl.
replace (C0 + a * C1 - a) with C0 by lca.
rewrite Cmod_0.
easy.
Qed.
Lemma limit_linear_poly : forall (a x : C),
limit_at_point (Peval [C0;a]) x (a * x).
Proof. unfold limit_at_point; intros.
destruct (Ceq_dec a C0); subst.
- exists 1.
split; try lra.
intros.
replace (([C0; C0]) [[x0]] - C0 * x) with C0 by lca.
rewrite Cmod_0.
easy.
- exists (ϵ / (Cmod a))%R.
split.
unfold Rdiv.
apply Rmult_gt_0_compat; auto.
apply Rinv_0_lt_compat.
apply Cmod_gt_0; easy.
intros.
replace (([C0; a]) [[x0]]) with (a * x0) by lca.
unfold Cminus.
rewrite Copp_mult_distr_r, <- Cmult_plus_distr_l, Cmod_mult.
assert (H2 : Cmod (x0 - x) * Cmod a < ϵ / Cmod a * Cmod a).
{ apply Rmult_lt_compat_r; auto.
apply Cmod_gt_0; easy. }
unfold Rdiv, Cminus in H2.
rewrite Rmult_assoc, Rinv_l in H2.
lra.
unfold not; intros.
apply n.
apply Cmod_eq_0; easy.
Qed.
Lemma limit_scale : forall (f : C -> C) (a c L : C),
limit_at_point f a L ->
limit_at_point (fun x => c * f x) a (c * L).
Proof. intros.
unfold limit_at_point in *.
intros.
destruct (Ceq_dec c 0); subst.
- exists 1.
split; try lra.
intros.
replace (C0 * f x - C0 * L) with C0 by lca.
rewrite Cmod_0; easy.
- destruct (H (ϵ / Cmod c)%R).
unfold Rdiv.
apply Rmult_lt_0_compat; auto.
apply Rinv_0_lt_compat.
apply Cmod_gt_0; auto.
exists (x)%R.
destruct H1.
split. auto.
intros.
unfold Cminus.
rewrite Copp_mult_distr_r, <- Cmult_plus_distr_l, Cmod_mult.
apply H2 in H3; auto.
apply (Rmult_lt_compat_l (Cmod c)) in H3.
unfold Rdiv in H3.
rewrite (Rmult_comm ϵ), <- Rmult_assoc, Rinv_r, Rmult_1_l in H3; auto.
unfold not; intros; apply n.
apply Cmod_eq_0; auto.
apply Cmod_gt_0; auto.
Qed.
Lemma limit_plus : forall (f1 f2 : C -> C) (a L1 L2 : C),
limit_at_point f1 a L1 -> limit_at_point f2 a L2 ->
limit_at_point (fun c => f1 c + f2 c) a (L1 + L2).
Proof. unfold limit_at_point.
intros.
assert (H2 : ϵ / 2 > 0).
{ lra. }
assert (H3 := H2).
apply H in H2.
apply H0 in H3.
destruct H2 as [δ1 [H2 H4] ]; destruct H3 as [δ2 [H3 H5] ].
exists (Rmin δ1 δ2); split.
apply Rmin_Rgt; auto.
intros.
assert (H8 : Cmod (f1 x - L1) < ϵ / 2).
{ apply H4; auto.
assert (H' := (Rmin_l δ1 δ2)); lra. }
assert (H9 : Cmod (f2 x - L2) < ϵ / 2).
{ apply H5; auto.
assert (H' := (Rmin_r δ1 δ2)); lra. }
apply (Rplus_lt_compat (Cmod (f1 x - L1)) (ϵ / 2) (Cmod (f2 x - L2)) (ϵ / 2)) in H8; auto.
replace (ϵ / 2 + ϵ / 2)%R with ϵ in H8 by lra.
replace (f1 x + f2 x - (L1 + L2))%C with ((f1 x - L1) + (f2 x - L2))%C by lca.
assert (H10 : Cmod (f1 x - L1 + (f2 x - L2)) <= Cmod (f1 x - L1) + Cmod (f2 x - L2)).
{ apply Cmod_triangle. }
lra.
Qed.
Lemma limit_mult_0 : forall (f1 f2 : C -> C) (a : C),
limit_at_point f1 a C0 -> limit_at_point f2 a C0 ->
limit_at_point (fun c => f1 c * f2 c) a C0.
Proof. intros.
unfold limit_at_point in *.
intros.
destruct (H (√ ϵ)); try apply sqrt_lt_R0; auto.
destruct (H0 (√ ϵ)); try apply sqrt_lt_R0; auto.
destruct H2; destruct H3.
exists (Rmin x x0).
split.
apply Rmin_pos; auto.
intros.
unfold Cminus in *.
rewrite Copp_0, Cplus_0_r in *.
rewrite Cmod_mult, <- (sqrt_def ϵ); try lra.
apply Rmult_le_0_lt_compat; try apply Cmod_ge_0.
all : rewrite <- (Cplus_0_r (_ x1)).
apply H4; auto.
assert (H8 := Rmin_l x x0). lra.
apply H5; auto.
assert (H8 := Rmin_r x x0). lra.
Qed.
Lemma limit_mult : forall (f1 f2 : C -> C) (a L1 L2 : C),
limit_at_point f1 a L1 -> limit_at_point f2 a L2 ->
limit_at_point (fun c => f1 c * f2 c) a (L1 * L2).
Proof. intros.
assert (H' : (fun c => f1 c * f2 c) =
(fun c => ((f1 c - L1) * (f2 c - L2)) + (L1 * f2 c) + (L2 * f1 c) - (L1 * L2))).
{ apply functional_extensionality.
intros. lca. }
replace (L1 * L2) with (C0 + L1 * L2 + L2 * L1 - L1 * L2) by lca.
rewrite H'.
repeat apply limit_plus.
unfold Cminus.
apply limit_mult_0.
replace C0 with (L1 + - L1) by lca.
apply limit_plus; auto.
apply limit_const.
replace C0 with (L2 + - L2) by lca.
apply limit_plus; auto.
apply limit_const.
all : try apply limit_scale; auto.
apply limit_const.
Qed.
Lemma continuous_plus : forall (f1 f2 : C -> C) (a : C),
continuous_at f1 a -> continuous_at f2 a ->
continuous_at (fun c => f1 c + f2 c) a.
Proof. intros.
unfold continuous_at in *.
apply (limit_plus f1 f2 a (f1 a) (f2 a)); easy.
Qed.
Lemma continuous_mult : forall (f1 f2 : C -> C) (a : C),
continuous_at f1 a -> continuous_at f2 a ->
continuous_at (fun c => f1 c * f2 c) a.
Proof. intros.
unfold continuous_at in *.
apply (limit_mult f1 f2 a (f1 a) (f2 a)); easy.
Qed.
Lemma constant_continuous_poly : forall (c a : C),
continuous_at (Peval [c]) a.
Proof. intros.
unfold continuous_at.
replace (([c]) [[a]]) with c by lca.
apply limit_const_poly.
Qed.
Lemma linear_continuous_poly : forall (c a : C),
continuous_at (Peval [C0; c]) a.
Proof. intros.
unfold continuous_at.
replace (([C0; c]) [[a]]) with (c * a) by lca.
apply limit_linear_poly.
Qed.
Lemma power_x_eval : forall (n : nat) (a x : C),
(repeat C0 n ++ [a]) [[x]] = a * x^n.
Proof. intros.
unfold Peval.
rewrite app_length; simpl.
rewrite Nat.add_1_r, <- big_sum_extend_r, <- (Nat.add_0_r (length (repeat C0 n))),
app_nth2_plus, Nat.add_0_r, repeat_length; simpl.
rewrite <- Cplus_0_l.
apply Cplus_simplify; auto.
apply (@big_sum_0_bounded C C_is_monoid); intros.
rewrite app_nth1.
destruct (nth_in_or_default x0 (repeat C0 n) C0).
- apply repeat_spec in i.
rewrite i; lca.
- rewrite e; lca.
- rewrite repeat_length; easy.
Qed.
Lemma power_x_mul_x : forall (n : nat) (a : C),
(repeat C0 (S n) ++ [a]) ≅ [C0; C1] *, (repeat C0 n ++ [a]).
Proof. intros.
simpl repeat.
rewrite <- app_comm_cons.
apply head_0_Pmult_x.
Qed.
Lemma continuous_power_x : forall (n : nat) (a b : C),
continuous_at (Peval ((repeat C0 n) ++ [b])) a.
Proof. induction n as [| n'].
- intros.
apply constant_continuous_poly.
- intros.
rewrite power_x_mul_x.
assert (H' : forall p q : Polynomial, Peval (p *, q) = fun c => p[[c]] * q[[c]]).
{ intros.
apply functional_extensionality.
intros.
apply Pmult_eval. }
rewrite H'.
apply continuous_mult.
apply linear_continuous_poly.
apply IHn'.
Qed.
Lemma polynomial_continuous : forall (p : Polynomial) (a : C),
(fun p => continuous_at (Peval p) a) p.
Proof. intros.
apply (@ind_from_end C).
- rewrite <- C0_Peq_nil.
apply constant_continuous_poly.
- intros.
rewrite app_eval.
apply continuous_plus.
apply H.
apply continuous_power_x.
Qed.
Lemma constant_ae_continuous : forall (f : C -> C) (a c : C),
continuous_at f c -> (forall x, x <> c -> f x = a) ->
f c = a.
Proof. intros.
unfold continuous_at in H.
assert (H1 : limit_at_point f c a).
{ unfold limit_at_point.
intros.
exists 1.
split; try lra.
intros.
rewrite H0; auto.
unfold Cminus.
rewrite Cplus_opp_r, Cmod_0.
lra. }
apply (limit_unique f c); easy.
Qed.
Lemma almost_0_implies_0 : forall (p : Polynomial),
(forall c, c <> C0 -> p[[c]] = C0) -> p ≅ [].
Proof. intros.
unfold Peq.
apply functional_extensionality; intros.
destruct (Ceq_dec x C0).
- replace (([]) [[x]]) with C0 by easy.
apply constant_ae_continuous.
apply polynomial_continuous.
intros.
apply H; subst; easy.
- apply H; easy.
Qed.
Lemma almost_Peq_implies_Peq : forall (p1 p2 : Polynomial),
(forall c, c <> C0 -> p1[[c]] = p2[[c]]) -> p1 ≅ p2.
Proof. intros.
assert (H' : p1 +, -, p2 ≅ []).
{ apply almost_0_implies_0.
intros.
unfold Popp.
rewrite Pplus_eval, Pmult_eval, H; auto.
replace (([- C1]) [[c]]) with (-C1) by lca.
lca. }
replace p2 with ([] +, p2) by easy.
rewrite <- H', Pplus_assoc, Pplus_opp_l, Pplus_0_r.
easy.
Qed.
Lemma Peq_head_eq : forall (p1 p2 : Polynomial) (a1 a2 : C),
(a1 :: p1) ≅ (a2 :: p2) -> a1 = a2.
Proof. intros.
unfold Peq, Peval in H.
apply (f_equal_inv C0) in H.
simpl length in H.
rewrite (big_sum_unique a1), (big_sum_unique a2) in H; auto.
all : exists O; split; try lia; split; try lca.
all : intros; destruct x'; try easy; lca.
Qed.
Lemma Peq_tail_Peq_helper : forall (p1 p2 : Polynomial),
(C0 :: p1) ≅ (C0 :: p2) -> p1 ≅ p2.
Proof. intros.
apply almost_Peq_implies_Peq; intros.
apply (f_equal_inv c) in H.
unfold Peval in *.
simpl length in *.
do 2 rewrite <- big_sum_extend_l in H.
simpl in H.
rewrite Cmult_0_l, Cplus_0_l, Cplus_0_l in H.
rewrite <- Cmult_1_l, <- (Cinv_l c), <- Cmult_assoc, (@big_sum_mult_l C _ _ _ C_is_ring); auto.
assert (H' : (fun x : nat => nth x p2 C0 * (c * c ^ x)) =
(fun x : nat => c * (nth x p2 C0 * c ^ x))).
{ apply functional_extensionality; intros.
lca. }
simpl in *.
rewrite <- H', <- H.
assert (H'' : (fun x : nat => nth x p1 C0 * (c * c ^ x)) =
(fun x : nat => c * (nth x p1 C0 * c ^ x))).
{ apply functional_extensionality; intros.
lca. }
rewrite H''.
rewrite <- (@big_sum_mult_l C _ _ _ C_is_ring), Cmult_assoc, Cinv_l; auto; lca.
Qed.
Lemma Peq_tail_Peq : forall (p1 p2 : Polynomial) (a1 a2 : C),
(a1 :: p1) ≅ (a2 :: p2) -> p1 ≅ p2.
Proof. intros.
assert (H' : a1 = a2).
{ apply Peq_head_eq in H; easy. }
rewrite cons_simplify, (cons_simplify p2) in H; subst.
assert (H'' : [-a2] +, [a2] +, (C0 :: p1) ≅ [-a2] +, [a2] +, (C0 :: p2)).
{ rewrite Pplus_assoc, H, <- Pplus_assoc.
easy. }
simpl in H''.
replace (- a2 + a2 + C0) with C0 in H'' by lca.
apply Peq_tail_Peq_helper in H''.
easy.
Qed.
Lemma Peq_app_Peq : forall (p p1 p2 : Polynomial),
(p ++ p1) ≅ (p ++ p2) -> p1 ≅ p2.
Proof. induction p as [| a]; try easy.
intros; simpl in H.
apply Peq_tail_Peq in H.
apply IHp; easy.
Qed.
Lemma Peq_length_eq_helper : forall (n : nat) (p1 p2 : Polynomial),
p1 ≅ p2 ->
length p1 = n -> length p2 = n ->
p1 = p2.
Proof. induction n as [| n'].
- intros.
destruct p1; destruct p2; try easy.
- intros.
destruct p1; destruct p2; try easy.
simpl in *.
apply Nat.succ_inj in H0.
apply Nat.succ_inj in H1.
repeat apply f_equal_gen; try easy.
apply (Peq_head_eq p1 p2); easy.
apply IHn'; try easy.
apply (Peq_tail_Peq _ _ c c0); easy.
Qed.
Lemma Peq_length_eq : forall (n : nat) (p1 p2 : Polynomial),
p1 ≅ p2 -> length p1 = length p2 ->
p1 = p2.
Proof. intros.
apply (Peq_length_eq_helper (length p1)); easy.
Qed.
Lemma Peq_nil_reduce : forall (p : Polynomial) (a : C),
(a :: p) ≅ [] -> a = C0 /\ p ≅ [].
Proof. intros.
rewrite <- C0_Peq_nil in H.
split.
- apply Peq_head_eq in H; easy.
- apply Peq_tail_Peq in H; easy.
Qed.
Lemma Peq_nil_contains_C0 : forall (p : Polynomial),
p ≅ [] -> (forall c, In c p -> c = C0).
Proof. induction p; try easy.
intros.
apply Peq_nil_reduce in H.
destruct H0; try (subst; easy).
apply IHp; easy.
Qed.
Lemma same_elem_same_rev : forall (p : Polynomial),
(forall c, In c p -> c = C0) -> p = repeat C0 (length p).
Proof. induction p as [| a]; try easy.
intros; simpl.
rewrite (H a), IHp, repeat_length; try easy.
intros.
apply H; right; easy.
left; easy.
Qed.
Lemma Peq_0_eq_repeat_0 : forall (p : Polynomial),
p ≅ [] -> p = repeat C0 (length p).
Proof. intros.
apply same_elem_same_rev.
apply Peq_nil_contains_C0; easy.
Qed.
Lemma Peq_nil_rev_Peq_nil : forall (p : Polynomial),
p ≅ [] -> rev p = p.
Proof. intros.
rewrite same_elem_same_rev, <- rev_repeat, <- same_elem_same_rev; try easy.
all : apply Peq_nil_contains_C0; easy.
Qed.
Lemma last_C0_Peq_front : forall (p : Polynomial),
p ++ [C0] ≅ p.
Proof. intros.
unfold Peq, Peval.
apply functional_extensionality; intros.
rewrite app_length; simpl.
rewrite Nat.add_1_r, <- big_sum_extend_r.
rewrite app_nth2, Nat.sub_diag; auto; simpl.
rewrite Cmult_0_l, Cplus_0_r.
apply big_sum_eq_bounded; intros.
rewrite app_nth1; easy.
Qed.