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Rings.v
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Require Import List.
Require Export Complex.
(*
Require Export Quantum.
Require Import FTA.
*)
(* how should I do this? *)
(*
Lemma Submonoid_theorem : forall (R : Set) `{Monoid R} (S : Set)
(S0 : S) (Splus : S -> S -> S) (StoR : S -> R),
(forall s1 s2, StoR (Splus s1 s2) = ((StoR s1) + (StoR s2))%G) ->
Monoid S.
Proof. intros.
split.
Lemma Subring_theorem : forall (R : Set) `{Comm_Ring R} (S : Set) (S0 S1 : S) (Splus : S -> S -> S)
(Sopp : S -> S) (Smult : S -> S -> S) (StoR : S -> R),
(forall s1 s2, StoR (Splus s1 s2) = ((StoR s1) + (StoR s2))%G) ->
(forall s, StoR (Sopp s) = (- (StoR s))%G) ->
Monoid S.
Proof. intros.
*)
Open Scope C_scope.
(* exponentiation by positives *)
Fixpoint Cpow_pos (c : C) (p : positive) : C :=
match p with
| xH => c
| xO p' => (Cpow_pos c p') * (Cpow_pos c p')
| xI p' => c * (Cpow_pos c p') * (Cpow_pos c p')
end.
Lemma Cpow_pos_to_nat : forall (c : C) (p : positive),
Cpow_pos c p = c ^ (Pos.to_nat p).
Proof. intros.
induction p.
- rewrite Pos2Nat.inj_xI, Nat.mul_comm; simpl.
rewrite Cpow_mult, IHp.
lca.
- rewrite Pos2Nat.inj_xO, Nat.mul_comm; simpl.
rewrite Cpow_mult, IHp.
lca.
- lca.
Qed.
Lemma Cpow_pos_0 : forall (p : positive), Cpow_pos C0 p = C0.
Proof.
intros p.
rewrite Cpow_pos_to_nat.
apply Cpow_0_l.
lia.
Qed.
Lemma Cpow_pos_nonzero : forall (c : C) (p : positive), c <> C0 -> Cpow_pos c p <> C0.
Proof. intros.
rewrite Cpow_pos_to_nat.
apply Cpow_nonzero.
easy.
Qed.
Lemma Cpow_pos_add_r : forall (c : C) (p1 p2 : positive),
Cpow_pos c (p1 + p2) = (Cpow_pos c p1) * (Cpow_pos c p2).
Proof. intros.
do 3 rewrite Cpow_pos_to_nat.
rewrite Pos2Nat.inj_add, Cpow_add.
easy.
Qed.
Lemma Cpow_pos_mult_r : forall (c : C) (p1 p2 : positive),
Cpow_pos c (p1 * p2) = Cpow_pos (Cpow_pos c p1) p2.
Proof. intros.
do 3 rewrite Cpow_pos_to_nat.
rewrite Pos2Nat.inj_mul, Cpow_mult.
easy.
Qed.
Lemma Cpow_pos_mult_l : forall (c1 c2 : C) (p : positive),
Cpow_pos (c1 * c2) p = (Cpow_pos c1 p) * (Cpow_pos c2 p).
Proof. intros.
do 3 rewrite Cpow_pos_to_nat.
rewrite Cpow_mul_l.
easy.
Qed.
Lemma Cpow_pos_succ : forall (c : C) (p : positive),
Cpow_pos c (Pos.succ p) = c * Cpow_pos c p.
Proof. intros.
induction p; simpl; auto.
- rewrite IHp; lca.
- lca.
Qed.
Lemma Cpow_pos_pred_double : forall (c : C) (p : positive),
Cpow_pos c (Pos.pred_double p) = (/ c) * (Cpow_pos c p * Cpow_pos c p).
Proof.
intros c p.
destruct (Ceq_dec c 0); [subst; rewrite Cpow_pos_0; lca|].
induction p; simpl; auto.
- repeat rewrite Cmult_assoc.
rewrite Cinv_l; try lca; auto.
- rewrite IHp.
repeat rewrite Cmult_assoc.
rewrite Cinv_r; try lca; auto.
- rewrite Cmult_assoc, Cinv_l; try lca; auto.
Qed.
Lemma Cpow_pos_inv : forall (c : C) (p : positive),
Cpow_pos (/ c) p = / (Cpow_pos c p).
Proof.
intros c p.
destruct (Ceq_dec c 0); [subst; rewrite Cinv_0, Cpow_pos_0; lca|].
induction p; simpl.
- rewrite IHp.
repeat rewrite Cinv_mult_distr; auto.
- rewrite IHp.
rewrite Cinv_mult_distr; try apply Cpow_pos_nonzero; auto.
- easy.
Qed.
Lemma Cpow_pos_real : forall (c : C) (p : positive),
snd c = 0 -> snd (Cpow_pos c p) = 0.
Proof. intros.
induction p; simpl.
- rewrite IHp, H; lra.
- rewrite IHp; lra.
- easy.
Qed.
Lemma Cpow_pos_1 : forall (p : positive),
Cpow_pos C1 p = C1.
Proof. intros.
induction p; simpl.
- rewrite IHp; lca.
- rewrite IHp; lca.
- lca.
Qed.
(* exponentiation by integers *)
Definition Cpow_int (c : C) (z : Z) : C :=
match z with
| Z0 => C1
| Zpos p => Cpow_pos c p
| Zneg p => / Cpow_pos c p
end.
Infix "^^" := Cpow_int (at level 10) : C_scope.
Lemma Cpow_int_nonzero : forall (c : C) (z : Z), c <> C0 -> c ^^ z <> C0.
Proof. intros.
destruct z.
- apply C1_neq_C0.
- apply Cpow_pos_nonzero; easy.
- apply nonzero_div_nonzero.
apply Cpow_pos_nonzero; easy.
Qed.
Lemma Cpow_int_0_l : forall (z : Z), z <> 0%Z -> C0 ^^ z = C0.
Proof.
intros z Hz.
destruct z; [easy|..];
simpl.
- apply Cpow_pos_0.
- rewrite Cpow_pos_0, Cinv_0.
reflexivity.
Qed.
Lemma Cpow_int_add_1 : forall (c : C) (z : Z),
c <> C0 -> c ^^ (1 + z) = c * c^^z.
Proof.
intros.
destruct z; try lca.
- destruct p; simpl; try lca.
rewrite Cpow_pos_succ; lca.
- destruct p; simpl.
+ rewrite <- Cmult_assoc, (Cinv_mult_distr c); auto.
destruct (Ceq_dec c 0);
[subst; now rewrite Cpow_pos_0, !Cmult_0_l, Cinv_0|].
rewrite Cmult_assoc, Cinv_r; try lca; auto.
+ rewrite Cpow_pos_pred_double, Cinv_mult_distr, Cinv_inv; auto.
+ rewrite Cinv_r; easy.
Qed.
Lemma Cpow_int_minus_1 : forall (c : C) (z : Z),
c <> C0 -> c ^^ (-1 + z) = / c * c^^z.
Proof. intros.
destruct z; try lca.
- destruct p; simpl; try lca.
+ repeat rewrite Cmult_assoc.
rewrite Cinv_l; auto; lca.
+ rewrite Cpow_pos_pred_double; auto.
+ rewrite Cinv_l; easy.
- destruct p; simpl; try lca.
+ rewrite Cpow_pos_succ, <- Cinv_mult_distr; auto.
apply f_equal; lca.
+ rewrite <- Cmult_assoc, Cinv_mult_distr; auto.
+ rewrite Cinv_mult_distr; easy.
Qed.
Lemma Cpow_int_add_nat : forall (c : C) (n : nat) (z : Z),
c <> C0 -> c ^^ (Z.of_nat n + z) = c^^(Z.of_nat n) * c^^z.
Proof. intros.
induction n; try lca.
replace (S n + z)%Z with (1 + (n + z))%Z by lia.
replace (Z.of_nat (S n)) with (1 + Z.of_nat n)%Z by lia.
repeat rewrite Cpow_int_add_1; auto.
rewrite IHn; lca.
Qed.
Lemma Cpow_int_minus_nat : forall (c : C) (n : nat) (z : Z),
c <> C0 -> c ^^ (- Z.of_nat n + z) = c^^(- Z.of_nat n) * c^^z.
Proof. intros.
induction n; try lca.
replace (- S n + z)%Z with (- 1 + (- n + z))%Z by lia.
replace (- (S n))%Z with (- 1 + - n)%Z by lia.
repeat rewrite Cpow_int_minus_1; auto.
rewrite IHn; lca.
Qed.
Theorem Cpow_int_add_r : forall (c : C) (z1 z2 : Z),
c <> C0 -> c ^^ (z1 + z2) = c^^z1 * c^^z2.
Proof. intros.
destruct (Z_plusminus_nat z1) as [n [H0 | H0]].
- rewrite H0, Cpow_int_add_nat; easy.
- rewrite H0, Cpow_int_minus_nat; easy.
Qed.
Lemma Cpow_int_mult_r : forall (c : C) (z1 z2 : Z),
c <> C0 -> c ^^ (z1 * z2) = (c ^^ z1) ^^ z2.
Proof. intros.
destruct z1; destruct z2; try lca; simpl.
all : try (rewrite Cpow_pos_1; lca).
all : rewrite Cpow_pos_mult_r; try lca.
all : rewrite Cpow_pos_inv; try apply Cpow_pos_nonzero; auto.
rewrite Cinv_inv; auto.
Qed.
Lemma Cpow_int_mult_l : forall (c1 c2 : C) (z : Z),
c1 <> C0 -> c2 <> C0 -> (c1 * c2) ^^ z = (c1 ^^ z) * (c2 ^^ z).
Proof. intros.
destruct z; try lca; simpl.
rewrite Cpow_pos_mult_l; easy.
rewrite Cpow_pos_mult_l, Cinv_mult_distr; auto;
apply Cpow_pos_nonzero; easy.
Qed.
Lemma Cpow_inv1 : forall (c : C) (z : Z), c <> C0 -> c^^(-z) = / (c^^z).
Proof. intros.
replace (-z)%Z with (z * -1)%Z by lia.
rewrite Cpow_int_mult_r; easy.
Qed.
Lemma Cpow_inv2 : forall (c : C) (z : Z), c <> C0 -> (/ c)^^z = / (c^^z).
Proof. intros.
replace z with (-1 * -z)%Z by lia.
rewrite Cpow_int_mult_r.
replace ((/ c) ^^ (-1)) with (/ / c) by easy.
replace (-1 * - z)%Z with z by lia.
rewrite Cinv_inv, Cpow_inv1; auto.
apply nonzero_div_nonzero; auto.
Qed.
(* checking that Cpow_int is consistent with Cpow on nats *)
Lemma Cpow_int_cons : forall (c : C) (n : nat),
c^n = c ^^ n.
Proof. intros.
destruct n; try lca.
unfold Cpow_int.
destruct (Z.of_nat (S n)) eqn:E; try easy.
rewrite Cpow_pos_to_nat.
apply f_equal_gen; try apply f_equal; auto.
apply Pos2Z.inj_pos in E.
rewrite <- E, SuccNat2Pos.id_succ.
easy.
Qed.
Lemma Cpow_int_real : forall (c : C) (z : Z),
c <> 0 -> snd c = 0 -> snd (c ^^ z) = 0.
Proof. intros.
destruct z; auto.
- simpl; apply Cpow_pos_real; auto.
- replace (Z.neg p) with (- Z.pos p)%Z by lia.
rewrite Cpow_inv1; auto.
apply div_real.
apply Cpow_pos_real; auto.
Qed.
(* foreboding: translating between Cpow_int and Zpow *)
Lemma ZtoC_pow_nat : forall (z : Z) (n : nat),
RtoC (IZR (z^n)%Z) = (RtoC (IZR z))^^n.
Proof. intros.
induction n; try easy.
rewrite <- Cpow_int_cons in *.
replace (S n) with (1 + n)%nat by lia.
rewrite Nat2Z.inj_add, Z.pow_add_r, Cpow_add_r; try lia.
rewrite mult_IZR, <- IHn, RtoC_mult, RtoC_pow, pow_IZR.
apply f_equal_gen; auto.
Qed.
(* foreboding: translating between Cpow_int and Zpow *)
Lemma ZtoC_pow : forall (z n : Z),
(n >= 0)%Z ->
RtoC (IZR (z^n)%Z) = (RtoC (IZR z))^^n.
Proof. intros.
destruct (Z_plusminus_nat n) as [x [H0 | H0]]; subst.
- rewrite ZtoC_pow_nat; easy.
- destruct x; try lia.
replace (-O)%Z with (Z.of_nat O) by lia.
rewrite ZtoC_pow_nat; easy.
Qed.
#[global] Hint Rewrite plus_IZR minus_IZR opp_IZR mult_IZR ZtoC_pow : ZtoR_db.
(* NB: use plus_IZR, mult_IZR, RtoC_plus, RtoC_mult to move between types: *)
(* quick helper tactic. TODO: centralize these *)
Ltac fastZtoC := repeat (autorewrite with ZtoR_db; autorewrite with RtoC_db; try lia).
(*
*
*
*)
(* fst d is the 2-adic value of d, snd d is (odd leftover part / 2) *)
(* should this be type or Set ?*)
(* TODO: could make Dpos Dneg, like Z, but this makes for a lot of cases *)
Inductive Dyadic : Set :=
| D0 : Dyadic
| Dn : Z -> Z -> Dyadic. (* fst arg = 2-adic value, snd arg = (odd_part / 2) *)
Definition D1 : Dyadic := Dn 0 0.
Definition D2 : Dyadic := Dn 1 0.
Definition Dhalf : Dyadic := Dn (Z.opp 1) 0.
Declare Scope D_scope.
Delimit Scope D_scope with D.
Open Scope D_scope.
Bind Scope D_scope with Dyadic.
Open Scope Z_scope.
Coercion IZR : Z >-> R.
Definition DtoC (x : Dyadic) : C :=
match x with
| D0 => C0
| Dn n x => (C2^^n * (C2 * x + 1))
end.
Coercion DtoC : Dyadic >-> C.
(* Coercions make things soo nice! *)
Lemma DtoC_D0 : DtoC D0 = C0.
Proof. easy. Qed.
Lemma DtoC_D1 : DtoC D1 = C1.
Proof. lca. Qed.
Lemma DtoC_D2 : DtoC D2 = C2.
Proof. lca. Qed.
Lemma DtoC_Dhalf : DtoC Dhalf = / C2.
Proof. unfold DtoC, Dhalf.
rewrite Cpow_inv1.
lca.
apply RtoC_neq.
lra.
Qed.
Lemma DtoC_neq_0 : forall (z z0 : Z),
(C2 ^^ z * (C2 * z0 + C1))%C <> C0.
Proof. intros.
apply Cmult_neq_0.
apply Cpow_int_nonzero.
apply RtoC_neq; lra.
replace (C2 * z0 + C1)%C with (RtoC (IZR (2 * z0 + 1)%Z)).
unfold not; intros.
apply RtoC_inj in H.
apply eq_IZR in H.
lia.
fastZtoC.
easy.
Qed.
Lemma move_power_2 : forall (z1 z2 : Z) (c1 c2 : C),
(C2 ^^ z1 * c1 = C2 ^^ z2 * c2)%C -> (c1 = C2 ^^ (z2 - z1) * c2)%C.
Proof. intros.
apply (Cmult_simplify (C2 ^^ (-z1)) (C2 ^^ (-z1))) in H; auto.
do 2 rewrite Cmult_assoc, <- Cpow_int_add_r in H.
replace (- z1 + z1) with 0 in H by lia; simpl in H.
rewrite Cmult_1_l in H; subst.
replace (- z1 + z2) with (z2 - z1) by lia.
easy.
all : apply RtoC_neq; lra.
Qed.
(* TODO: move these next two lemmas somewhere else, they are never used here *)
Lemma log2_inj_0 : forall (n : nat),
(C2 ^ n)%C = C1 -> n = O.
Proof. intros.
destruct n; try easy; simpl in *.
assert (H' : forall n, (C2 ^ n)%C = RtoC (IZR ((2 ^ n)%Z))).
{ induction n0; auto.
replace (C2 ^ S n0)%C with (C2 * (C2 ^ n0))%C by easy.
rewrite IHn0.
autorewrite with ZtoR_db; try lia.
replace (Z.of_nat (S n0)) with (1 + n0)%Z by lia.
rewrite Cpow_int_add_r; try easy.
apply RtoC_neq; lra. }
rewrite H' in H.
simpl in H.
rewrite <- RtoC_mult in H.
apply RtoC_inj in H.
rewrite <- mult_IZR in H.
apply eq_IZR in H.
lia.
Qed.
Lemma log2_inj : forall (n1 n2 : nat),
Cpow C2 n1 = Cpow C2 n2 -> n1 = n2.
Proof. induction n1.
- intros.
replace (C2 ^ 0)%C with C1 in H by easy.
symmetry in H.
apply log2_inj_0 in H; easy.
- intros.
destruct n2.
replace (C2 ^ 0)%C with C1 in H by easy.
apply log2_inj_0 in H; easy.
simpl in H.
rewrite (IHn1 n2); auto.
apply Cmult_cancel_l in H; auto.
apply RtoC_neq; lra.
Qed.
Lemma DtoC_inj_uneq_case : forall z z0 z1 z2 : Z,
z < z1 ->
(C2 ^^ z * (C2 * z0 + C1))%C = (C2 ^^ z1 * (C2 * z2 + C1))%C ->
False.
Proof. intros.
apply move_power_2 in H0.
assert (H' : 0 < z1 - z). lia.
destruct (Z_plusminus_nat (z1 - z)) as [x [H1 | H1]]; try lia.
rewrite H1 in *.
destruct x; try easy.
rewrite <- ZtoC_pow_nat in H0.
repeat rewrite <- RtoC_mult, <- mult_IZR in H0.
repeat rewrite <- RtoC_plus, <- plus_IZR in H0.
repeat rewrite <- RtoC_mult, <- mult_IZR in H0.
apply RtoC_inj in H0.
apply eq_IZR in H0.
replace (Z.of_nat (S x)) with (1 + x)%Z in H0 by lia.
rewrite Z.pow_add_r in H0; try lia.
Qed.
(* idea: WLOG v(x) <= v(y), equality case is easy, then reflect back to Z *)
Lemma DtoC_inj : forall (x y : Dyadic),
DtoC x = DtoC y -> x = y.
Proof.
intros.
unfold DtoC in *.
destruct x; destruct y; try easy.
- assert (H' := DtoC_neq_0 z z0).
rewrite H in H'; easy.
- assert (H' := DtoC_neq_0 z z0).
rewrite H in H'; easy.
- destruct (Ztrichotomy_inf z z1) as [[H0 | H0] | H0].
+ apply DtoC_inj_uneq_case in H; lia.
+ subst.
apply move_power_2 in H.
replace (z1 - z1)%Z with 0%Z in H by lia.
rewrite Cmult_1_l in H.
apply (Cplus_simplify _ _ (-C1) (-C1)) in H; auto.
replace (C2 * z0 + C1 + - C1)%C with (C2 * z0)%C in H by lca.
replace (C2 * z2 + C1 + - C1)%C with (C2 * z2)%C in H by lca.
apply Cmult_cancel_l in H.
apply RtoC_inj in H.
apply eq_IZR in H; subst; easy.
apply RtoC_neq; lra.
+ symmetry in H.
apply DtoC_inj_uneq_case in H; lia.
Qed.
(* this is actually overkill because Z is decible, but shows usefulness of DtoC_inj *)
Lemma Deq_dec (d1 d2 : Dyadic) : { d1 = d2 } + { ~ (d1 = d2) }.
Proof.
destruct (Ceq_dec (DtoC d1) (DtoC d2)).
- apply DtoC_inj in e.
left; easy.
- right.
unfold not; intros; apply n.
rewrite H; easy.
Qed.
(* TODO: create 'lda', which is like lca but for dyadics *)
(*
(* lca, a great tactic for solving computations or basic equality checking *)
Lemma c_proj_eq : forall (c1 c2 : C), fst c1 = fst c2 -> snd c1 = snd c2 -> c1 = c2.
Proof. intros c1 c2 H1 H2. destruct c1, c2. simpl in *. subst. reflexivity. Qed.
(* essentially, we just bootsrap coq's lra *)
Ltac lca := eapply c_proj_eq; simpl; lra.
*)
(** *** Arithmetic operations *)
(* this one will be terrible... but just need to prove DtoC_plus! *)
Definition Dplus (x y : Dyadic) : Dyadic :=
match (x, y) with
| (D0, _) => y
| (_, D0) => x
| (Dn n x', Dn m y') =>
match Ztrichotomy_inf n m with
| inleft (left _) => (* x has lower power of 2 *)
Dn n (2^(m-n-1) * (2*y'+1) + x')
| inleft (right _) => (* equal power of 2 *)
match odd_part ((2*x'+1) + (2*y'+1)) with
| Z0 => D0
| _ => Dn (n + two_val ((2*x'+1) + (2*y'+1)))
(((odd_part ((2*x'+1) + (2*y'+1))) - 1) / 2)
end (* y has lower power of 2 *)
| inright _ => Dn m (2^(n-m-1) * (2*x'+1) + y')
end
end.
Definition Dopp (x : Dyadic) : Dyadic :=
match x with
| D0 => D0
| Dn n x => Dn n (-x-1)
end.
Definition Dminus (x y : Dyadic) : Dyadic := Dplus x (Dopp y).
(* this one is much easier! *)
Definition Dmult (x y : Dyadic) : Dyadic :=
match (x, y) with
| (D0, _) => D0
| (_, D0) => D0
| (Dn n x, Dn m y) => Dn (n + m) (2*x*y + x + y)
end.
Fixpoint Dpow (d : Dyadic) (n : nat) : Dyadic :=
match n with
| 0%nat => D1
| S n' => Dmult d (Dpow d n')
end.
Infix "+," := Dplus (at level 50, left associativity) : D_scope.
Infix "*," := Dmult (at level 40, left associativity) : D_scope.
Notation "-, d" := (Dopp d) (at level 35) : D_scope.
Infix "-," := Dminus (at level 50, left associativity) : D_scope.
Infix "^," := Dpow (at level 30) : D_scope.
(* showing compatability *)
(* could make this shorter, but it was also expected that this would be the hardest part,
so its messiness is warrented *)
Lemma DtoC_plus : forall d1 d2, DtoC (d1 +, d2) = (DtoC d1 + DtoC d2)%C.
Proof. intros.
destruct d1; destruct d2; simpl; try lca.
unfold Dplus; destruct (Ztrichotomy_inf z z1); try destruct s.
- unfold Dplus, DtoC.
fastZtoC.
rewrite (Cmult_plus_distr_l _ _ z0), (Cmult_assoc C2).
replace (C2 * C2 ^^ (z1 - z - 1))%C with (C2 ^^ (z1 - z)).
rewrite <- Cplus_assoc, (Cmult_plus_distr_l _ _ (C2 * z0 + C1)), Cmult_assoc.
replace (C2 ^^ z * C2 ^^ (z1 - z))%C with (C2 ^^ z1).
lca.
rewrite <- Cpow_int_add_r.
apply f_equal; lia.
apply RtoC_neq; lra.
replace (Cmult C2) with (Cmult (C2 ^^ 1)) by (apply f_equal; easy).
rewrite <- Cpow_int_add_r.
apply f_equal; lia.
apply RtoC_neq; lra.
- subst.
replace (2 * z0 + 1 + (2 * z2 + 1))%Z with (2 * (z0 + z2 + 1))%Z by lia.
rewrite odd_part_reduce.
destruct (Z.eq_dec (z0 + z2 + 1) Z0).
rewrite e. simpl.
replace (C2 ^^ z1 * (C2 * z0 + C1) + C2 ^^ z1 * (C2 * z2 + C1))%C with
(C2 ^^ z1 * C2 * (z0 + z2 + C1))%C by lca.
do 2 rewrite <- RtoC_plus, <- plus_IZR.
rewrite e.
lca.
assert (H' : odd_part (z0 + z2 + 1) <> 0).
unfold not; intros; apply n.
apply odd_part_0; auto.
destruct (odd_part (z0 + z2 + 1)) eqn:E; try easy.
all : unfold DtoC;
repeat (repeat rewrite <- RtoC_plus, <- plus_IZR;
repeat rewrite <- RtoC_mult, <- mult_IZR).
all : rewrite <- E, odd_part_odd, Cpow_int_add_r, <- Cmult_plus_distr_l,
<- Cmult_assoc; auto; try apply f_equal.
all : try (apply RtoC_neq; lra).
all : rewrite <- ZtoC_pow, <- odd_part_reduce; try apply two_val_ge_0.
all : rewrite <- RtoC_mult, <- mult_IZR, <- RtoC_plus, <- plus_IZR.
all : rewrite <- twoadic_breakdown; try lia.
all : fastZtoC; lca.
- unfold Dplus, DtoC.
fastZtoC.
rewrite (Cmult_plus_distr_l _ _ z2), (Cmult_assoc C2).
replace (C2 * C2 ^^ (z - z1 - 1))%C with (C2 ^^ (z - z1)).
rewrite <- Cplus_assoc, (Cmult_plus_distr_l _ _ (C2 * z2 + C1)), Cmult_assoc.
replace (C2 ^^ z1 * C2 ^^ (z - z1))%C with (C2 ^^ z).
lca.
rewrite <- Cpow_int_add_r.
apply f_equal; lia.
apply RtoC_neq; lra.
replace (Cmult C2) with (Cmult (C2 ^^ 1)) by (apply f_equal; easy).
rewrite <- Cpow_int_add_r.
apply f_equal; lia.
apply RtoC_neq; lra.
Qed.
Lemma DtoC_opp : forall d, DtoC (-, d) = (- (DtoC d))%C.
Proof. intros.
destruct d; try lca.
unfold Dopp, DtoC, Zminus.
fastZtoC.
rewrite Copp_mult_distr_r.
lca.
Qed.
Lemma DtoC_minus : forall d1 d2, DtoC (d1 -, d2) = ((DtoC d1) - (DtoC d2))%C.
Proof. intros.
unfold Cminus, Dminus.
rewrite DtoC_plus, DtoC_opp.
easy.
Qed.
(* this is surprisingly easy! *)
Lemma DtoC_mult : forall d1 d2, DtoC (d1 *, d2) = (DtoC d1 * DtoC d2)%C.
Proof. intros.
destruct d1; destruct d2; try lca.
unfold Dmult, DtoC.
fastZtoC.
rewrite Cpow_int_add_r.
lca.
apply RtoC_neq; lra.
Qed.
Lemma DtoC_pow : forall d n, DtoC (d ^, n) = ((DtoC d) ^ n)%C.
Proof. intros.
induction n.
- lca.
- simpl.
rewrite DtoC_mult, IHn.
easy.
Qed.
#[global] Hint Rewrite DtoC_plus DtoC_opp DtoC_minus DtoC_mult DtoC_pow : DtoC_db.
Ltac lDa_try1 := apply DtoC_inj; autorewrite with DtoC_db; lca.
(* Note that once we have a injective homomorphism into C,
we get all the ring axioms for free! *)
(* TODO: generalize this in Summation.v *)
Lemma Dplus_comm : forall d1 d2, d1 +, d2 = d2 +, d1.
Proof. intros. lDa_try1. Qed.
Lemma Dplus_assoc : forall d1 d2 d3, d1 +, (d2 +, d3) = d1 +, d2 +, d3.
Proof. intros. lDa_try1. Qed.
Lemma Dplus_0_r : forall d, d +, D0 = d.
Proof. intros. lDa_try1. Qed.
Lemma Dplus_0_l : forall d, D0 +, d = d.
Proof. intros. lDa_try1. Qed.
Lemma Dminus_0_r : forall d, d -, D0 = d.
Proof. intros. lDa_try1. Qed.
Lemma Dminus_0_l : forall d, D0 -, d = -, d.
Proof. intros. lDa_try1. Qed.
Lemma Dmult_0_r : forall d, d *, D0 = D0.
Proof. intros. lDa_try1. Qed.
Lemma Dmult_0_l : forall d, D0 *, d = D0.
Proof. intros. lDa_try1. Qed.
Lemma Dmult_comm : forall d1 d2, d1 *, d2 = d2 *, d1.
Proof. intros. lDa_try1. Qed.
Lemma Dmult_assoc : forall d1 d2 d3, d1 *, (d2 *, d3) = d1 *, d2 *, d3.
Proof. intros. lDa_try1. Qed.
Lemma Dmult_1_r : forall d, d *, D1 = d.
Proof. intros. lDa_try1. Qed.
Lemma Dmult_1_l : forall d, D1 *, d = d.
Proof. intros. lDa_try1. Qed.
Lemma Dopp_mult_distr_l : forall d1 d2, -, (d1 *, d2) = (-, d1) *, d2.
Proof. intros. lDa_try1. Qed.
Lemma Dopp_mult_distr_r : forall d1 d2, -, (d1 *, d2) = d1 *, (-, d2).
Proof. intros. lDa_try1. Qed.
Lemma Dopp_mult_neg_l : forall d, -, d = (-, D1) *, d.
Proof. intros. lDa_try1. Qed.
Lemma Dopp_mult_neg_r : forall d, -, d = d *, (-, D1).
Proof. intros. lDa_try1. Qed.
Lemma Dminus_diag : forall d1 d2 : Dyadic, d1 = d2 -> d1 -, d2 = D0.
Proof. intros; subst.
apply DtoC_inj.
rewrite DtoC_minus.
replace (DtoC D0) with C0 by lca.
apply Cminus_diag.
easy.
Qed.
Lemma Dminus_eq_0 : forall r1 r2 : Dyadic, r1 -, r2 = D0 -> r1 = r2.
Proof. intros.
unfold Dminus, Dopp, Dplus.
apply DtoC_inj.
apply Cminus_eq_0.
rewrite <- DtoC_minus, H.
easy.
Qed.
Lemma Dmult_neq_0 : forall d1 d2 : Dyadic, d1 <> D0 -> d2 <> D0 -> d1 *, d2 <> D0.
Proof. intros.
unfold not; intros.
apply (f_equal_gen DtoC DtoC) in H1; auto.
rewrite DtoC_mult, DtoC_D0 in H1.
apply Cmult_integral in H1.
destruct H1.
apply H; apply DtoC_inj; easy.
apply H0; apply DtoC_inj; easy.
Qed.
Lemma Dmult_integral : forall d1 d2 : Dyadic, d1 *, d2 = D0 -> d1 = D0 \/ d2 = D0.
Proof. intros.
destruct (Deq_dec d1 D0); try (left; easy).
destruct (Deq_dec d2 D0); try (right; easy).
apply (Dmult_neq_0 d1 d2) in n0; auto.
easy.
Qed.
Lemma Dpow_add : forall (d : Dyadic) (n m : nat), d ^, (n + m) = d^,n *, d^,m.
Proof. intros.
apply DtoC_inj.
rewrite DtoC_mult.
do 3 rewrite DtoC_pow.
rewrite Cpow_add; easy.
Qed.
Lemma Dpow_mult : forall (d : Dyadic) (n m : nat), d ^, (n * m) = (d ^, n) ^, m.
Proof. intros.
apply DtoC_inj.
do 3 rewrite DtoC_pow.
rewrite Cpow_mult; easy.
Qed.
(* we showcase two different approaches to numerical proofs based on the above *)
Lemma Dhalf_double_1 : Dhalf +, Dhalf = D1.
Proof. unfold Dhalf, Dplus; simpl.
replace (0 / 2) with 0%Z.
easy.
rewrite Zdiv.Zdiv_0_l.
easy.
Qed.
Lemma Dhalf_double_2 : Dhalf +, Dhalf = D1.
Proof. apply DtoC_inj.
rewrite DtoC_plus.
rewrite DtoC_Dhalf, DtoC_D1.
lca.
Qed.
(* I prefer the second way *)
Lemma Dmult_2_half : D2 *, Dhalf = D1.
Proof. apply DtoC_inj.
rewrite DtoC_mult.
rewrite DtoC_Dhalf, DtoC_D2, DtoC_D1.
lca.
Qed.
Lemma Dmult_half_2 : Dhalf *, D2 = D1.
Proof. apply DtoC_inj.
rewrite DtoC_mult.
rewrite DtoC_Dhalf, DtoC_D2, DtoC_D1.
lca.
Qed.
Lemma Dpow_2_pos : forall (n : positive), D2 ^, (Pos.to_nat n) = Dn (Z.pos n) 0.
Proof. intros.
apply DtoC_inj.
rewrite DtoC_pow; simpl.
rewrite Cpow_pos_to_nat, Cmult_0_r, Cplus_0_l, Cmult_1_r, Cmult_1_r.
easy.
Qed.
Lemma Dpow_2_nat : forall (n : nat), D2 ^, (S n) = Dn (Z.of_nat (S n)) 0.
Proof. intros; simpl.
rewrite <- Dpow_2_pos.
replace (D2 *, D2 ^, n) with (D2 ^, (S n)) by easy.
apply f_equal_gen; auto.
rewrite SuccNat2Pos.id_succ; easy.
Qed.
(* some more specific lemmas: *)
Lemma Dyadic_real : forall (d : Dyadic), snd (DtoC d) = 0.
Proof. intros.
destruct d; unfold DtoC; simpl; try lra.
rewrite Cpow_int_real; simpl; try lra.
apply RtoC_neq; lra.
Qed.
Lemma Cmult_d_l : forall (d : Dyadic) (r1 r2 : R),
(d * (r1, r2))%C = (fst (DtoC d) * r1, fst (DtoC d) * r2)%R.
Proof. intros.
unfold Cmult, DtoC; destruct d; simpl.
all : apply injective_projections; simpl; try lra.
all : replace (snd (C2 ^^ z)) with 0%R; try lra.
all : rewrite Cpow_int_real; auto.
all : apply C0_fst_neq; simpl; lra.
Qed.
Lemma Cconj_D : forall (d : Dyadic), Cconj d = d.
Proof. intros.
destruct (DtoC d) eqn:E.
unfold Cconj; simpl.
replace r0 with (snd (DtoC d)) by (rewrite E; easy).
rewrite Dyadic_real.
apply c_proj_eq; simpl; lra.
Qed.
Lemma Dyadic_is_0 : forall (d : Dyadic),
fst (DtoC d) = 0 ->
d = D0.
Proof. intros.
apply DtoC_inj.
apply injective_projections; auto.
do 2 rewrite Dyadic_real.
easy.
Qed.
(*
*
*
*)
(* now we add Z *)
Close Scope Z_scope.
Open Scope D_scope.
(* the nonzero case is quite adhoc here, might want to use more machinery from above *)
Definition ZtoD (z : Z) : Dyadic :=
match z with
| Z0 => D0
| Zpos p => Dn (two_val z) ((odd_part z - 1) / 2)
| Zneg p => Dn (two_val z) ((odd_part z - 1) / 2)
end.
Lemma ZtoD_0 : ZtoD 0 = D0.
Proof. easy. Qed.
Lemma ZtoD_1 : ZtoD 1 = D1.
Proof. easy. Qed.
Lemma ZtoD_2 : ZtoD 2 = D2.
Proof. easy. Qed.
Lemma ZtoDtoC_is_ZtoRtoC : forall (z : Z), DtoC (ZtoD z) = RtoC (IZR z).
Proof. intros.
destruct z; try lca;
destruct p;
unfold ZtoD, DtoC;
rewrite <- RtoC_mult, <- mult_IZR, <- RtoC_plus, <- plus_IZR, odd_part_odd;
try lia; rewrite <- ZtoC_pow; try apply two_val_ge_0;
rewrite <- RtoC_mult, <- mult_IZR, <- twoadic_breakdown; try lia;
easy.
Qed.
(*
(*TODO: deal with this warning *)
(* This should really be Z -> D -> Q -> R -> C *)
Coercion ZtoD : Z >-> Dyadic.
*)
(* we now see the power of the injections and ZtoDtoC_is_ZtoRtoC for these next lemmas ! *)
(* TODO: automate proofs like this? *)