Require Import Arith Lia List Bool.

From Undecidability.Shared.Libs.DLW.Utils
 Require Import utils_tac gcd sums rel_iter bool_nat power_decomp prime.

Set Implicit Arguments.

Set Default Proof Using "Type".

Local Notation power := (mscal mult 1).
Local Notation "∑" := (msum plus 0).
Local Infix "≲" := binary_le (at level 70, no associativity).
Local Infix "⇣" := nat_meet (at level 40, left associativity).
Local Infix "⇡" := nat_join (at level 50, left associativity).

Hint Resolve power2_gt_0 : core.

Section stability_of_power.

 Fact mult_lt_power_2 u v k : u < power k 2 -> v < power k 2 -> u *v < power (2*k) 2.
 Proof.
 intros H1 H2.
 replace (2*k) with (k+k) by lia.
 rewrite power_plus.
 apply lt_le_trans with ((S u)*S v).
 simpl; rewrite (mult_comm _ (S _)); simpl; rewrite mult_comm; lia.
 apply mult_le_compat; auto.
 Qed.

 Fact mult_lt_power_2_4 u v k : u < power k 2 -> v < power k 2 -> u *v < power (4*k) 2.
 Proof.
 intros H1 H2.
 apply lt_le_trans with (1 := mult_lt_power_2 _ H1 H2).
 apply power_mono_l; lia.
 Qed.

 Fact mult_lt_power_2_4' u1 v1 u2 v2 k :
 u1 < power k 2
 -> v1 < power k 2
 -> u2 < power k 2
 -> v2 < power k 2
 -> u1 *v1 +v2 *u2 < power (4*k) 2.
 Proof.
 intros H1 H2 H3 H4.
 destruct (eq_nat_dec k 0) as [ ? | Hk ].
 - subst k; simpl.
 rewrite power_0 in *.
 destruct u1; destruct v1; destruct u2; destruct v2; subst; lia.
 - apply lt_le_trans with (power (S (2*k)) 2).
 + rewrite power_S, <- mult_2_eq_plus.
 apply plus_lt_compat; apply mult_lt_power_2; auto.
 + apply power_mono_l; lia.
 Qed.

End stability_of_power.

Section power_decomp.

 Variable (p : nat) (Hp : 2 <= p).

 Let power_nzero x : power x p <> 0.
 Proof. generalize (@power_ge_1 x p); lia. Qed.

 Fact power_decomp_lt n f a q :
 (forall i j, i < j < n -> f i < f j )
 -> (forall i, i < n -> f i < q )
 -> (forall i, i < n -> a i ∑ n (fun i => a i * power (f i ) p ) < power q p.
 Proof using Hp.
 revert q; induction n as [ | n IHn ]; intros q Hf1 Hf2 Ha.
 + rewrite msum_0; apply power_ge_1; lia.
 + rewrite msum_plus1; auto.
 apply lt_le_trans with (1*power (f n) p + a n * power (f n) p).
 * apply plus_lt_le_compat; auto.
 rewrite Nat.mul_1_l.
 apply IHn.
 - intros; apply Hf1; lia.
 - intros; apply Hf1; lia.
 - intros; apply Ha; lia.
 * rewrite <- Nat.mul_add_distr_r.
 replace q with (S (q-1)).
 - rewrite power_S; apply mult_le_compat; auto.
 ++ apply Ha; auto.
 ++ apply power_mono_l; try lia.
 generalize (Hf2 n); intros; lia.
 - generalize (Hf2 0); intros; lia.
 Qed.

 Lemma power_decomp_is_digit n a f :
 (forall i j, i < j < n -> f i < f j )
 -> (forall i, i < n -> a i forall i, i < n -> is_digit (∑ n (fun i => a i * power (f i ) p )) p (f i) (a i).
 Proof using Hp.
 intros Hf Ha.
 induction n as [ | n IHn ]; intros i Hi.
 + lia.
 + split; auto.
 exists (∑ (n -i ) (fun j => a (S i + j ) * power (f (S i +j ) - f i - 1) p )),
 (∑ i (fun j => a j * power (f j ) p )); split.
 - replace (S n) with (S i + (n-i)) by lia.
 rewrite msum_plus, msum_plus1; auto.
 rewrite <- plus_assoc, plus_comm; f_equal.
 rewrite Nat.mul_add_distr_r, plus_comm; f_equal.
 rewrite <- mult_assoc, mult_comm, <- sum_0n_scal_l.
 apply msum_ext.
 intros j Hj.
 rewrite (mult_comm (_ * _));
 repeat rewrite <- mult_assoc; f_equal.
 rewrite <- power_S, <- power_plus; f_equal.
 generalize (Hf i (S i+j)); intros; lia.
 - apply power_decomp_lt; auto.
 * intros; apply Hf; lia.
 * intros; apply Ha; lia.
 Qed.

 Theorem power_decomp_unique n f a b :
 (forall i j, i < j < n -> f i < f j )
 -> (forall i, i < n -> a i (forall i, i < n -> b i ∑ n (fun i => a i * power (f i ) p )
 = ∑ n (fun i => b i * power (f i ) p )
 -> forall i, i < n -> a i = b i.
 Proof using Hp.
 intros Hf Ha Hb E i Hi.
 generalize (power_decomp_is_digit _ _ Hf Ha Hi)
 (power_decomp_is_digit _ _ Hf Hb Hi).
 rewrite E; apply is_digit_fun.
 Qed.

End power_decomp.

Section power_decomp_uniq.

 Variable (p : nat) (Hp : 2 <= p).

 Theorem power_decomp_factor n f a :
 (forall i, 0 f 0 < f i )
 -> ∑ (S n ) (fun i => a i * power (f i ) p )
 = ∑ n (fun i => a (S i ) * power (f (S i ) - f 0 - 1) p ) * power (S (f 0)) p
 + a 0 * power (f 0) p.
 Proof using Hp.
 intros Hf.
 rewrite msum_S, plus_comm; f_equal.
 rewrite <- sum_0n_scal_r.
 apply msum_ext.
 intros i Hi.
 rewrite <- mult_assoc; f_equal.
 rewrite <- power_plus; f_equal.
 generalize (Hf (S i)); intros; lia.
 Qed.

 Let power_nzero x : power x p <> 0.
 Proof.
 generalize (@power_ge_1 x p); lia.
 Qed.

 Let lt_minus_cancel a b c : a b - a - 1 < c - a - 1.
 Proof. intros; lia. Qed.

 Theorem power_decomp_unique' n f a b :
 (forall i j, i < j < n -> f i < f j )
 -> (forall i, i < n -> a i (forall i, i < n -> b i ∑ n (fun i => a i * power (f i ) p )
 = ∑ n (fun i => b i * power (f i ) p )
 -> forall i, i < n -> a i = b i.
 Proof using Hp.
 revert f a b.
 induction n as [ | n IHn ]; intros f a b Hf Ha Hb.
 + intros; lia.
 + assert (forall i, 0 f 0 < f i)
 by (intros; apply Hf; lia).
 do 2 (rewrite power_decomp_factor; auto).
 intros E.
 apply div_rem_uniq in E; auto.
 * destruct E as (E1 & E2).
 intros [ | i ] Hi.
 - revert E2; rewrite Nat.mul_cancel_r; auto.
 - apply IHn with (4 := E1); try lia.
 ++ intros u j Hu; apply lt_minus_cancel; split; apply Hf; lia.
 ++ intros; apply Ha; lia.
 ++ intros; apply Hb; lia.
 * rewrite power_S.
 apply Nat.mul_lt_mono_pos_r.
 - apply power_ge_1; lia.
 - apply Ha; lia.
 * rewrite power_S.
 apply Nat.mul_lt_mono_pos_r.
 - apply power_ge_1; lia.
 - apply Hb; lia.
 Qed.

End power_decomp_uniq.

Fact mult_2_eq_plus x : x + x = 2 *x.
Proof. ring. Qed.

Section power_injective.

 Local Lemma power_2_inj_1 i j n : j 2* power n 2 <> power i 2 + power j 2.
 Proof.
 rewrite <- power_S; intros H4 E.
 generalize (@power_ge_1 j 2); intro C.
 destruct (lt_eq_lt_dec i (S n)) as [ [ H5 | H5 ] | H5 ].
 + apply power_mono_l with (x := 2) in H5; auto.
 rewrite power_S in H5.
 apply power_mono_l with (x := 2) in H4; auto.
 rewrite power_S in H4; lia.
 + subst i; lia.
 + apply power_mono_l with (x := 2) in H5; auto.
 rewrite power_S in H5; lia.
 Qed.

 Fact power_2_n_ij_neq i j n : i <> j -> power (S n) 2 <> power i 2 + power j 2.
 Proof.
 intros H.
 destruct (lt_eq_lt_dec i j) as [ [] | ]; try tauto.
 + rewrite plus_comm; apply power_2_inj_1; auto.
 + apply power_2_inj_1; auto.
 Qed.

 Fact power_2_inj i j : power i 2 = power j 2 -> i = j.
 Proof.
 intros H.
 destruct (lt_eq_lt_dec i j) as [ [ C | C ] | C ]; auto;
 apply power_smono_l with (x := 2) in C; lia.
 Qed.

 Local Lemma power_plus_lt a b c : a power a 2 + power b 2 < power c 2.
 Proof.
 intros [ H1 H2 ].
 apply power_mono_l with (x := 2) in H2; auto.
 apply power_smono_l with (x := 2) in H1; auto.
 rewrite power_S in H2; lia.
 Qed.

 Local Lemma power_inj_2 i1 j1 i2 j2 :
 j1 < i1
 -> j2 < i2
 -> power i1 2 + power j1 2 = power i2 2 + power j2 2
 -> i1 = i2 /\ j1 = j2.
 Proof.
 intros H1 H2 H3.
 destruct (lt_eq_lt_dec i1 i2) as [ [ C | C ] | C ].
 + generalize (@power_plus_lt j1 i1 i2); intros; lia.
 + split; auto; apply power_2_inj; subst; lia.
 + generalize (@power_plus_lt j2 i2 i1); intros; lia.
 Qed.

 Theorem sum_2_power_2_injective i1 j1 i2 j2 :
 j1 <= i1
 -> j2 <= i2
 -> power i1 2 + power j1 2 = power i2 2 + power j2 2
 -> i1 = i2 /\ j1 = j2.
 Proof.
 intros H1 H2 E.
 destruct (eq_nat_dec i1 j1) as [ H3 | H3 ];
 destruct (eq_nat_dec i2 j2) as [ H4 | H4 ].
 + subst j1 j2.
 assert (i1 = i2); auto.
 do 2 rewrite mult_2_eq_plus, <- power_S in E.
 apply power_2_inj in E; lia.
 + subst j1; rewrite mult_2_eq_plus in E.
 apply power_2_inj_1 in E; lia.
 + subst j2; symmetry in E.
 rewrite mult_2_eq_plus in E.
 apply power_2_inj_1 in E; lia.
 + revert E; apply power_inj_2; lia.
 Qed.

End power_injective.

Fact divides_power p a b : a <= b -> divides (power a p) (power b p).
Proof.
 * induction 1 as [ | b H IH ].
 + apply divides_refl.
 + apply divides_trans with (1 := IH).
 rewrite power_S; apply divides_mult, divides_refl.
Qed.

Fact divides_msum k n f : (forall i, i < n -> divides k (f i)) -> divides k (∑ n f).
Proof.
 revert f; induction n as [ | n IHn ]; intros f Hf.
 + rewrite msum_0; apply divides_0.
 + rewrite msum_S; apply divides_plus.
 * apply Hf; lia.
 * apply IHn; intros; apply Hf; lia.
Qed.

Fact inc_seq_split_lt n f k :
 (forall i j, i < j < n -> f i < f j )
 -> { p | p <= n /\ (forall i, i f i < k ) /\ forall i, p <= i < n -> k <= f i }.
Proof.
 revert f; induction n as [ | n IHn ]; intros f Hf.
 + exists 0; split; auto; split; intros; lia.
 + destruct (le_lt_dec k (f 0)) as [ H | H ].
 - exists 0; split; try lia.
 split; intros i Hi; try lia.
 destruct i as [ | i ]; auto.
 apply le_trans with (1 := H), lt_le_weak, Hf; lia.
 - destruct (IHn (fun i => f (S i))) as (p & H1 & H2 & H3).
 * intros; apply Hf; lia.
 * exists (S p); split; try lia; split.
 ++ intros [ | i ] Hi; auto; apply H2; lia.
 ++ intros [ | i ] Hi; try lia; apply H3; lia.
Qed.

Fact inc_seq_split_le n f h : (forall i j, i < j < n -> f i < f j )
 -> { q | q <= n
 /\ (forall i, i < q -> f i <= h )
 /\ (forall i, q <= i < n -> h < f i ) }.
Proof.
 intros Hf.
 destruct inc_seq_split_lt with (1 := Hf) (k := S h)
 as (q & H1 & H2 & H3); exists q; split; auto; split.
 + intros i Hi; specialize (H2 _ Hi); lia.
 + intros i Hi; specialize (H3 _ Hi); lia.
Qed.

Fact divides_lt p q : q divides p q -> q = 0.
Proof.
 intros H1 ([ | k] & H2); auto.
 revert H2; simpl; generalize (k *p); intros; lia.
Qed.

Fact sum_powers_inc_lt_last n f r :
 2 <= r
 -> (forall i j, i < j <= n -> f i < f j )
 -> ∑ (S n ) (fun i => power (f i ) r ) < power (S (f n)) r.
Proof.
 intros Hr.
 revert f.
 induction n as [ | n IHn ]; intros f Hf.
 + rewrite msum_1; auto; apply power_smono_l; auto.
 + rewrite msum_plus1; auto.
 rewrite power_S.
 apply lt_le_trans with (power (S (f n)) r + power (f (S n)) r).
 * apply plus_lt_compat_r; auto.
 apply IHn; intros; apply Hf; lia.
 * assert (power (S (f n)) r <= power (f (S n)) r) as H.
 { apply power_mono_l; try lia; apply Hf; lia. }
 apply le_trans with (2 * power (f (S n)) r); try lia.
 apply mult_le_compat; auto.
Qed.

Fact sum_powers_inc_lt n f p r :
 2 <= r
 -> (forall i, i < n -> f i (forall i j, i < j < n -> f i < f j )
 -> ∑ n (fun i => power (f i ) r ) < power p r.
Proof.
 destruct n as [ | n ].
 + intros H _ _; rewrite msum_0; apply power_ge_1; lia.
 + intros H1 H2 H3.
 apply lt_le_trans with (power (S (f n)) r).
 * apply sum_powers_inc_lt_last; auto.
 intros; apply H3; lia.
 * apply power_mono_l; try lia.
 apply H2; auto.
Qed.

Fact sum_powers_injective r n f m g :
 2 <= r
 -> (forall i j, i < j < n -> f i < f j )
 -> (forall i j, i < j < m -> g i < g j )
 -> ∑ n (fun i => power (f i ) r ) = ∑ m (fun i => power (g i ) r )
 -> n = m /\ forall i, i < n -> f i = g i.
Proof.
 intros Hr; revert m f g.
 induction n as [ | n IHn ]; intros m f g Hf Hg.
 + rewrite msum_0.
 destruct m as [ | m ].
 * rewrite msum_0; split; auto; intros; lia.
 * rewrite msum_S.
 generalize (@power_ge_1 (g 0) r); intros; exfalso; lia.
 + destruct m as [ | m ].
 * rewrite msum_0, msum_S; intros; exfalso.
 generalize (@power_ge_1 (f 0) r); intros; exfalso; lia.
 * destruct (lt_eq_lt_dec (f n) (g m)) as [ [E|E]| E].
 - rewrite msum_plus1 with (n := m); auto.
 intros; exfalso.
 assert (∑ (S n ) (fun i => power (f i ) r ) < power (g m) r) as C; try lia.
 apply sum_powers_inc_lt; auto.
 intros i Hi.
 destruct (eq_nat_dec i n); subst; auto.
 apply lt_trans with (2 := E), Hf; lia.
 - do 2 (rewrite msum_plus1; auto); intros C.
 destruct (IHn m f g) as (H1 & H2).
 ++ intros; apply Hf; lia.
 ++ intros; apply Hg; lia.
 ++ rewrite E in C; lia.
 ++ split; subst; auto.
 intros i Hi.
 destruct (eq_nat_dec i m); subst; auto.
 apply H2; lia.
 - rewrite msum_plus1 with (n := n); auto.
 intros; exfalso.
 assert (∑ (S m ) (fun i => power (g i ) r ) < power (f n) r) as C; try lia.
 apply sum_powers_inc_lt; auto.
 intros i Hi.
 destruct (eq_nat_dec i m); subst; auto.
 apply lt_trans with (2 := E), Hg; lia.
Qed.

Fact power_divides_sum_power r p n f :
 2 <= r
 -> 0 < n
 -> (forall i j, i < j < n -> f i < f j )
 -> divides (power p r) (∑ n (fun i => power (f i ) r )) <-> p <= f 0.
Proof.
 intros Hr Hn Hf.
 split.
 + destruct inc_seq_split_lt with (k := p) (1 := Hf) as (k & H1 & H2 & H3).
 replace n with (k+(n-k)) by lia.
 rewrite msum_plus; auto.
 rewrite plus_comm; intros H.
 apply divides_plus_inv in H.
 2: apply divides_msum; intros; apply divides_power, H3; lia.
 destruct k as [ | k ].
 * apply H3; lia.
 * apply divides_lt in H.
 - rewrite msum_S in H.
 generalize (@power_ge_1 (f 0) r); intros; lia.
 - apply sum_powers_inc_lt; auto.
 intros; apply Hf; lia.
 + intros H.
 apply divides_msum.
 intros i Hi; apply divides_power.
 apply le_trans with (1 := H).
 destruct i; auto.
 generalize (Hf 0 (S i)); intros; lia.
Qed.

Fact smono_upto_injective n f :
 (forall i j, i < j < n -> f i < f j )
 -> (forall i j, i < n -> j < n -> f i = f j -> i = j ).
Proof.
 intros Hf i j Hi Hj E.
 destruct (lt_eq_lt_dec i j) as [ [H|] | H ]; auto.
 + generalize (@Hf i j); intros; lia.
 + generalize (@Hf j i); intros; lia.
Qed.

Fact product_sums n f g : (∑ n f )*(∑ n g )
 = ∑ n (fun i => f i *g i )
 + ∑ n (fun i => ∑ i (fun j => f i *g j + f j *g i )).
Proof.
 induction n as [ | n IHn ].
 + repeat rewrite msum_0; auto.
 + repeat rewrite msum_plus1; auto.
 repeat rewrite Nat.mul_add_distr_l.
 repeat rewrite Nat.mul_add_distr_r.
 rewrite IHn, msum_sum; auto.
 * rewrite sum_0n_scal_l, sum_0n_scal_r; ring.
 * intros; ring.
Qed.

Section sums.

 Fact square_sum n f : (∑ n f )*(∑ n f ) = ∑ n (fun i => f i *f i ) + 2*∑ n (fun i => ∑ i (fun j => f i *f j )).
 Proof.
 rewrite product_sums, <- sum_0n_scal_l; f_equal.
 apply msum_ext; intros; rewrite <- sum_0n_scal_l.
 apply msum_ext; intros; ring.
 Qed.

 Fact sum_regroup r k n f :
 (forall i, i < n -> f i < k )
 -> (forall i j, i < j < n -> f i < f j )
 -> { g | ∑ n (fun i => power (f i ) r )
 = ∑ k (fun i => g i * power i r )
 /\ (forall i, i < k -> g i <= 1)
 /\ (forall i, k <= i -> g i = 0) }.
 Proof.
 revert k f; induction n as [ | n IHn ]; intros k f Hf1 Hf2.
 + exists (fun _ => 0); split; auto.
 rewrite msum_0, msum_of_unit; auto.
 + destruct (IHn (f n) f) as (g & H1 & H2 & H3).
 * intros; apply Hf2; lia.
 * intros; apply Hf2; lia.
 * exists (fun i => if eq_nat_dec i (f n) then 1 else g i).
 split; [ | split ].
 - rewrite msum_plus1, H1; auto.
 replace k with (f n + S (k - f n -1)).
 2: generalize (Hf1 n); intros; lia.
 rewrite msum_plus; auto; f_equal.
 ++ apply msum_ext.
 intros i He.
 destruct (eq_nat_dec i (f n)); try ring; lia.
 ++ rewrite msum_S, msum_of_unit; auto.
 ** repeat (rewrite plus_comm; simpl).
 destruct (eq_nat_dec (f n) (f n)); try ring; lia.
 ** intros i Hi.
 destruct (eq_nat_dec (f n+S i) (f n)); try lia.
 rewrite H3; lia.
 - intros i Hi.
 destruct (eq_nat_dec i (f n)); auto.
 destruct (le_lt_dec (f n) i).
 ++ rewrite H3; lia.
 ++ apply H2; lia.
 - intros i Hi.
 generalize (Hf1 n); intros.
 destruct (eq_nat_dec i (f n)); try lia.
 apply H3; lia.
 Qed.

 Section sum_sum_regroup.

 Variable (r n k : nat) (f : nat -> nat)
 (Hf1 : forall i, i < n -> f i <= k)
 (Hf2 : forall i j, i < j < n -> f i < f j).

 Theorem sum_sum_regroup : { g | ∑ n (fun i => ∑ i (fun j => power (f i + f j ) r ))
 = ∑ (2*k ) (fun i => g i * power i r )
 /\ forall i, g i <= n }.
 Proof using Hf1 Hf2.
 revert n f Hf1 Hf2.
 induction n as [ | p IHp ]; intros f Hf1 Hf2.
 + exists (fun _ => 0); split; auto.
 rewrite msum_0.
 simpl; rewrite msum_of_unit; auto.
 + destruct (IHp f) as (g & H1 & H2).
 * intros; apply Hf1; lia.
 * intros; apply Hf2; lia.
 * destruct sum_regroup with (r := r) (n := p) (f := fun j => f p + f j) (k := 2*k)
 as (g1 & G1 & G2 & G3).
 - intros i Hi; generalize (@Hf1 p) (@Hf2 i p); intros; lia.
 - intros i j H; generalize (@Hf2 i j); intros; lia.
 - assert (forall i, g1 i <= 1) as G4.
 { intro i; destruct (le_lt_dec (2*k) i); auto; rewrite G3; lia. }
 exists (fun i => g i + g1 i); split.
 ++ rewrite msum_plus1; auto.
 rewrite H1, G1, <- msum_sum; auto.
 2: intros; ring.
 apply msum_ext; intros; ring.
 ++ intros i.
 generalize (H2 i) (G4 i); intros; lia.
 Qed.

 End sum_sum_regroup.

 Section all_ones.

 Local Lemma equation_inj x y a b : 1 <= x -> 1+x *a = y -> 1+x *b = y -> a = b.
 Proof.
 intros H1 H2 H3.
 rewrite <- H3 in H2; clear y H3.
 rewrite <- (@Nat.mul_cancel_l _ _ x); lia.
 Qed.

 Variables (r : nat) (Hr : 2 <= r).

 Fact all_ones_equation l : 1+(r -1)*∑ l (fun i => power i r ) = power l r.
 Proof using Hr.
 induction l as [ | l IHl ].
 * rewrite msum_0, Nat.mul_0_r, power_0; auto.
 * rewrite msum_plus1; auto.
 rewrite Nat.mul_add_distr_l, power_S.
 replace r with (1+(r -1)) at 4 by lia.
 rewrite Nat.mul_add_distr_r.
 rewrite <- IHl at 2; ring.
 Qed.

 Fact all_ones_dio l w : w = ∑ l (fun i => power i r ) <-> 1+(r -1)*w = power l r.
 Proof using Hr.
 split.
 + intros; subst; apply all_ones_equation.
 + intros H.
 apply equation_inj with (2 := H).
 * lia.
 * apply all_ones_equation.
 Qed.

 End all_ones.

 Section const_1.

 Variable (l q : nat) (Hl : 0 < l) (Hlq : l +1 < q).

 Let Hq : 1 <= q. Proof. lia. Qed.
 Let Hq' : 0 < 4*q. Proof. lia. Qed.

 Let r := (power (4*q) 2).

 Let Hr' : 4 <= r. Proof. apply (@power_mono_l 2 (4*q) 2); lia. Qed.
 Let Hr : 2 <= r. Proof. lia. Qed.

 Section all_ones.

 Variable (n w : nat) (Hw : w = ∑ n (fun i => power i r )).

 Local Lemma Hw_0 : w = ∑ n (fun i => 1*power i r ).
 Proof using Hw. rewrite Hw; apply msum_ext; intros; ring. Qed.

 Fact all_ones_joins : w = msum nat_join 0 n (fun i => 1*power i r).
 Proof using Hl Hlq Hw.
 rewrite Hw_0.
 apply sum_powers_ortho with (q := 4*q); auto; try lia.
 Qed.

 Local Lemma Hw_1 : 2*w = ∑ n (fun i => 2*power i r ).
 Proof using Hw.
 rewrite Hw_0, <- sum_0n_scal_l.
 apply msum_ext; intros; ring.
 Qed.

 Fact all_ones_2_joins : 2*w = msum nat_join 0 n (fun i => 2*power i r).
 Proof using Hl Hlq Hw.
 rewrite Hw_1.
 apply sum_powers_ortho with (q := 4*q); auto; try lia.
 Qed.

 End all_ones.

 Section increase.

 Variable (m k k' u w : nat) (f : nat -> nat)
 (Hm : 2*m < r)
 (Hf1 : forall i, i < m -> f i <= k)
 (Hf2 : forall i j, i < j < m -> f i < f j)
 (Hw : w = ∑ k' (fun i => power i r ))
 (Hu : u = ∑ m (fun i => power (f i ) r )).

 Let Hf4 : forall i j, i < m -> j < m -> f i = f j -> i = j.
 Proof. apply smono_upto_injective; auto. Qed.

 Let u1 := ∑ m (fun i => power (2*f i ) r ).
 Let u2 := ∑ m (fun i => ∑ i (fun j => 2*power (f i + f j ) r )).

 Fact const_u_square : u * u = u1 + u2.
 Proof using Hl Hlq Hw Hu Hm.
 unfold u1, u2.
 rewrite Hu, square_sum; f_equal.
 + apply msum_ext; intros; rewrite <- power_plus; f_equal; lia.
 + rewrite <- sum_0n_scal_l; apply msum_ext; intros i Hi.
 rewrite <- sum_0n_scal_l; apply msum_ext; intros j Hj.
 rewrite power_plus; ring.
 Qed.

 Local Lemma Hu1_0 : u1 = ∑ m (fun i => 1*power (2*f i ) r ).
 Proof. apply msum_ext; intros; ring. Qed.

 Local Lemma Hseq_u a : a <= m -> ∑ a (fun i => 1*power (2*f i ) r ) = msum nat_join 0 a (fun i => 1*power (2*f i) r).
 Proof using Hw Hf2 Hl Hlq Hm Hu.
 intros Ha.
 apply sum_powers_ortho with (q := 4*q); auto; try lia.
 intros i j Hi Hj ?; apply Hf4; lia.
 Qed.

 Local Lemma Hu1 : u1 = msum nat_join 0 m (fun i => 1*power (2*f i) r).
 Proof using Hw Hf2 Hl Hlq Hm Hu.
 rewrite Hu1_0; apply Hseq_u; auto.
 Qed.

 Local Lemma Hu2_0 : u2 = 2 * ∑ m (fun i => ∑ i (fun j => power (f i + f j ) r )).
 Proof.
 unfold u2; rewrite <- sum_0n_scal_l; apply msum_ext.
 intros; rewrite <- sum_0n_scal_l; apply msum_ext; auto.
 Qed.

 Local Lemma g_full : { g | ∑ m (fun i => ∑ i (fun j => power (f i + f j ) r ))
 = ∑ (2*k ) (fun i : nat => g i * power i r )
 /\ forall i : nat, g i <= m }.
 Proof using Hf1 Hf2. apply sum_sum_regroup; auto. Qed.

 Let g := proj1_sig g_full.
 Local Lemma Hg1 : u2 = ∑ (2*k ) (fun i => (2*g i ) * power i r ).
 Proof.
 rewrite Hu2_0, (proj1 (proj2_sig g_full)), <- sum_0n_scal_l.
 apply msum_ext; unfold g; intros; ring.
 Qed.

 Local Lemma Hg2 i : 2*g i <= 2*m.
 Proof. apply mult_le_compat; auto; apply (proj2_sig g_full). Qed.

 Let Hg3 i : 2*g i < r.
 Proof using Hm. apply le_lt_trans with (1 := Hg2 _); auto. Qed.

 Let Hu2 : u2 = msum nat_join 0 (2*k) (fun i => (2*g i ) * power i r).
 Proof.
 rewrite Hg1.
 apply sum_powers_ortho with (q := 4*q); auto; lia.
 Qed.

 Let Hu1_u2_1 : u1 ⇣ u2 = 0.
 Proof.
 rewrite Hu1, Hu2.
 apply nat_ortho_joins.
 intros i j Hi Hj.
 destruct (eq_nat_dec j (2*f i)) as [ H | H ].
 + unfold r; do 2 rewrite <- power_mult.
 rewrite <- H.
 rewrite nat_meet_mult_power2.
 rewrite nat_meet_12n; auto.
 + rewrite nat_meet_powers_neq with (q := 4*q); auto; lia.
 Qed.

 Let Hu1_u2 : u *u = u1 ⇡ u2.
 Proof.
 rewrite const_u_square.
 apply nat_ortho_plus_join; auto.
 Qed.

 Let Hw_1 : w = msum nat_join 0 k' (fun i => 1*power i r).
 Proof. rewrite Hw; apply all_ones_joins; auto. Qed.

 Let H2w_1 : 2*w = msum nat_join 0 k' (fun i => 2*power i r).
 Proof. rewrite Hw; apply all_ones_2_joins; auto. Qed.

 Local Lemma Hu2_w : u2 ⇣ w = 0.
 Proof using Hf1 Hf2 H2w_1 Hu1_u2.
 rewrite Hu2, Hw_1.
 destruct (le_lt_dec k' (2*k)) as [ Hk | Hk ].
 2: { apply nat_ortho_joins.
 intros i j Hi Hj.
 rewrite nat_meet_comm.
 destruct (eq_nat_dec i j) as [ H | H ].
 + subst j; rewrite nat_meet_powers_eq with (q := 4*q); auto.
 rewrite nat_meet_12n; auto.
 + apply nat_meet_powers_neq with (q := 4*q); auto; try lia. }
 replace (2*k) with (k'+(2*k -k')) by lia.
 rewrite msum_plus, nat_meet_comm, nat_meet_join_distr_l, nat_join_comm; auto.
 rewrite (proj2 (nat_ortho_joins k' (2*k -k') _ _)), nat_join_0n.
 2: { intros i j H1 H2.
 apply nat_meet_powers_neq with (q := 4*q); auto; try lia. }
 apply nat_ortho_joins.
 intros i j Hi Hj.
 destruct (eq_nat_dec i j) as [ H | H ].
 + subst j; rewrite nat_meet_powers_eq with (q := 4*q); auto.
 rewrite nat_meet_12n; auto.
 + apply nat_meet_powers_neq with (q := 4*q); auto; try lia.
 Qed.

 Fact const_u1_prefix : { q | q <= m /\ u *u ⇣ w = ∑ q (fun i => 1*power (2*f i ) r ) }.
 Proof using H2w_1 Hf1 Hf2 Hu1_u2.
 destruct inc_seq_split_lt with (n := m) (f := fun i => 2*f i) (k := k') as (a & H1 & H2 & H3).
 + intros i j Hij; apply Hf2 in Hij; lia.
 + exists a; split; auto.
 rewrite Hu1_u2, nat_meet_comm, nat_meet_join_distr_l.
 do 2 rewrite (nat_meet_comm w).
 rewrite Hu2_w, nat_join_n0.
 rewrite Hu1, Hw_1.
 replace m with (a+(m -a)) by lia.
 rewrite msum_plus, nat_meet_comm, nat_meet_join_distr_l.
 rewrite nat_join_comm.
 rewrite (proj2 (nat_ortho_joins k' (m -a) _ _)), nat_join_0n; auto.
 3: apply nat_join_monoid.
 * rewrite Hseq_u; auto.
 rewrite nat_meet_comm.
 apply binary_le_nat_meet.
 apply nat_joins_binary_le.
 intros i Hi.
 exists (2*f i); split; auto.
 * intros; apply nat_meet_powers_neq with (q := 4*q); auto; try lia.
 generalize (H3 (a + j)); intros; lia.
 Qed.

 Hypothesis (Hk : 2*k < k').

 Let Hu1_w : u1 ⇣ w = u1.
 Proof.
 apply binary_le_nat_meet.
 rewrite Hu1, Hw_1.
 apply nat_joins_binary_le.
 intros i Hi.
 exists (2*f i); split; auto.
 apply le_lt_trans with (2 := Hk), mult_le_compat; auto.
 Qed.

 Let Hu1_2w : u1 ⇣ (2*w ) = 0.
 Proof.
 rewrite H2w_1, Hu1, nat_ortho_joins.
 intros i j Hi Hj.
 destruct (eq_nat_dec j (2 * f i)) as [ H | H ].
 + rewrite <- H, nat_meet_powers_eq with (q := 4*q); auto; try lia.
 rewrite nat_meet_12; auto.
 + apply nat_meet_powers_neq with (q := 4*q); auto; try lia.
 Qed.

 Fact const_u1_meet p : p = (u *u ) ⇣ w <-> p = u1.
 Proof using Hu1_w.
 rewrite Hu1_u2, nat_meet_comm, nat_meet_join_distr_l.
 do 2 rewrite (nat_meet_comm w).
 rewrite Hu1_w, Hu2_w, nat_join_n0; tauto.
 Qed.

 Fact const_u1_eq : (u *u ) ⇣ w = u1.
 Proof using Hu1_w. apply const_u1_meet; auto. Qed.

 Hypothesis Hf : forall i, i < m -> f i = power (S i) 2.

 Let Hu2_1 : u2 = msum nat_join 0 m (fun i => msum nat_join 0 i (fun j => 2*power (f i + f j) r)).
 Proof.
 unfold u2.
 apply double_sum_powers_ortho with (q := 4*q); auto; try lia.
 intros ? ? ? ? ? ?; repeat rewrite Hf; try lia.
 intros E.
 apply sum_2_power_2_injective in E; lia.
 Qed.

 Let Hu2_2w : u2 ⇣ (2*w ) = u2.
 Proof.
 apply binary_le_nat_meet.
 rewrite H2w_1, Hu2_1.
 apply nat_double_joins_binary_le.
 intros i j Hij.
 exists (f i + f j); split; auto.
 apply le_lt_trans with (2*f i); auto.
 + apply Hf2 in Hij; lia.
 + apply le_lt_trans with (2 := Hk), mult_le_compat; auto.
 apply Hf1; lia.
 Qed.

 Fact const_u2_meet p : p = (u *u ) ⇣ (2*w ) <-> p = u2.
 Proof using Hu1_w Hu2_2w.
 rewrite Hu1_u2, nat_meet_comm, nat_meet_join_distr_l.
 do 2 rewrite (nat_meet_comm (2*w)).
 rewrite Hu1_2w, Hu2_2w, nat_join_0n; tauto.
 Qed.

 End increase.

 Let Hl'' : 2*l < r.
 Proof.
 unfold r.
 rewrite (mult_comm _ q), power_mult.
 change (power 4 2) with 16.
 apply power_smono_l with (x := 16) in Hlq; try lia.
 apply le_lt_trans with (2 := Hlq).
 rewrite plus_comm; simpl plus; rewrite power_S.
 apply mult_le_compat; try lia.
 apply power_ge_n; lia.
 Qed.

 Section const_1_cn.

 Variable (u u1 : nat) (Hu : u = ∑ l (fun i => power (power (S i ) 2) r ))
 (Hu1 : u1 = ∑ l (fun i => power (power (S (S i )) 2) r )).

 Let w := ∑ (S (power (S l ) 2)) (fun i => power i r ).
 Let u2 := ∑ l (fun i => ∑ i (fun j => 2*power (power (S i ) 2 + power (S j ) 2) r )).

 Let H18 : 1+(r -1)*w = power (S (power (S l) 2)) r.
 Proof. rewrite <- all_ones_dio; auto. Qed.

 Let H19 : u *u = u1 + u2.
 Proof.
 rewrite Hu1, Hu.
 apply const_u_square with (k' := 0) (w := 0); eauto.
 Qed.

 Let k := S (power (S l) 2).
 Let f i := power (S i) 2.

 Let Hf1 i : i < l -> 2*f i < k.
 Proof.
 unfold k, f.
 intros; rewrite <- power_S; apply le_n_S, power_mono_l; lia.
 Qed.

 Let Hf2 i j : i < j < l -> f i < f j.
 Proof. intros; apply power_smono_l; lia. Qed.

 Let Hf3 i1 j1 i2 j2 : j1 <= i1 < l -> j2 <= i2 < l -> f i1 + f j1 = f i2 + f j2 -> i1 = i2 /\ j1 = j2.
 Proof.
 unfold f; intros H1 H2 E.
 apply sum_2_power_2_injective in E; lia.
 Qed.

 Let H20 : u1 = (u *u ) ⇣ w.
 Proof.
 rewrite const_u1_meet with (k := power l 2) (m := l) (f := f); auto.
 * intros i Hi; specialize (Hf1 Hi).
 revert Hf1; unfold k; rewrite power_S; intros; lia.
 * rewrite <- power_S; auto.
 Qed.

 Let H21 : u2 = (u *u ) ⇣ (2*w ).
 Proof.
 rewrite const_u2_meet with (k := power l 2) (m := l) (f := f); auto.
 * intros i Hi; specialize (Hf1 Hi).
 revert Hf1; unfold k; rewrite power_S; intros; lia.
 * rewrite <- power_S; auto.
 Qed.

 Let H22 : power 2 r + u1 = u + power (power (S l) 2) r.
 Proof.
 rewrite Hu, Hu1.
 destruct l.
 + do 2 rewrite msum_0.
 rewrite power_1; auto.
 + rewrite msum_plus1, msum_S; auto.
 rewrite power_1; ring.
 Qed.

 Let H23 : divides (power 4 r) u1.
 Proof.
 rewrite Hu1.
 apply divides_msum.
 intros i _.
 apply divides_power.
 apply (@power_mono_l 2 _ 2); lia.
 Qed.

 Lemma const1_cn : exists w u2 , 1+(r -1)*w = power (S (power (S l) 2)) r
 /\ u *u = u1 + u2
 /\ u1 = (u *u ) ⇣ w
 /\ u2 = (u *u ) ⇣ (2*w )
 /\ power 2 r + u1 = u + power (power (S l) 2) r
 /\ divides (power 4 r) u1.
 Proof using Hu1 Hu Hr'.
 exists w, u2; repeat (split; auto).
 Qed.

 End const_1_cn.

 Section const_1_cs.

 Variable (w u u1 u2 : nat).

 Hypothesis (H18 : 1+(r -1)*w = power (S (power (S l) 2)) r)
 (H19 : u *u = u1 + u2)
 (H20 : u1 = (u *u ) ⇣ w)
 (H21 : u2 = (u *u ) ⇣ (2*w ))
 (H22 : power 2 r + u1 = u + power (power (S l) 2) r)
 (H23 : divides (power 4 r) u1).

 Let Hw_0 : w = ∑ (S (power (S l ) 2)) (fun i => power i r ).
 Proof. apply all_ones_dio; auto. Qed.

 Let Hw_1 : w = ∑ (S (power (S l ) 2)) (fun i => 1*power i r ).
 Proof. rewrite Hw_0; apply msum_ext; intros; ring. Qed.

 Let Hw : w = msum nat_join 0 (S (power (S l) 2)) (fun i => 1*power i r).
 Proof. apply all_ones_joins; auto. Qed.

 Let H2w : 2*w = msum nat_join 0 (S (power (S l) 2)) (fun i => 2*power i r).
 Proof. apply all_ones_2_joins; auto. Qed.

 Let Hu1_0 : u1 ≲ ∑ (S (power (S l ) 2)) (fun i => 1*power i r ).
 Proof. rewrite H20, <- Hw_1; auto. Qed.

 Local Lemma mk_full : { m : nat & { k | u1 = ∑ (S m ) (fun i => power (k i ) r )
 /\ m <= power (S l) 2
 /\ (forall i, i < S m -> k i <= power (S l) 2)
 /\ forall i j, i < j < S m -> k i < k j } }.
 Proof using Hw Hu1_0 H19 H21 H22.
 assert ({ k : nat &
 { g : nat -> nat &
 { h | u1 = ∑ k (fun i => g i * power (h i ) r )
 /\ k <= S (power (S l) 2)
 /\ (forall i, i < k -> g i <> 0 /\ g i ≲ 1)
 /\ (forall i, i < k -> h i < S (power (S l) 2))
 /\ (forall i j, i < j < k -> h i < h j ) } } }) as H.
 { apply (@sum_powers_binary_le_inv _ Hq' r eq_refl _ (fun _ => _) (fun i => i)); auto.
 intros; lia. }
 destruct H as (m' & g & h & H1 & H2 & H3 & H4 & H5).
 assert (H6 : forall i, i < m' -> g i = 1).
 { intros i Hi; generalize (H3 _ Hi).
 intros (? & G2); apply binary_le_le in G2; lia. }
 assert (H7 : u1 = ∑ m' (fun i => 1 * power (h i ) r )).
 { rewrite H1; apply msum_ext; intros; rewrite H6; try ring; lia. }
 assert (H8 : u1 = ∑ m' (fun i => power (h i ) r )).
 { rewrite H7; apply msum_ext; intros; ring. }
 assert (H9 : m' <> 0).
 { intros E; rewrite E, msum_0 in H1.
 assert (power 2 r < power (power (S l) 2) r) as C.
 { apply power_smono_l; auto.
 apply (@power_smono_l 1 _ 2); lia. }
 lia. }
 destruct m' as [ | m ]; try lia.
 exists m, h; repeat (split; auto).
 + lia.
 + intros i; generalize (H4 i); intros; lia.
 Qed.

 Let m := projT1 mk_full.
 Let k := proj1_sig (projT2 mk_full).

 Let Hu1 : u1 = ∑ (S m ) (fun i => power (k i ) r ). Proof. apply (proj2_sig (projT2 mk_full)). Qed.
 Let Hm : m <= (power (S l) 2). Proof. apply (proj2_sig (projT2 mk_full)). Qed.
 Let Hk1 : forall i, i < S m -> k i <= power (S l) 2. Proof. apply (proj2_sig (projT2 mk_full)). Qed.
 Let Hk2 : forall i j, i < j < S m -> k i < k j. Proof. apply (proj2_sig (projT2 mk_full)). Qed.

 Let Hh_0 : 4 <= k 0.
 Proof.
 rewrite Hu1 in H23.
 apply power_divides_sum_power in H23; auto; try lia.
 Qed.

 Let f1 i := match i with 0 => 2 | S i => k i end.
 Let f2 i := if le_lt_dec i m then power (S l) 2 else k i.

 Let Hf1_0 : forall i, i <= S m -> f1 i < S (power (S l) 2).
 Proof.
 intros [ | i ] Hi; simpl; apply le_n_S.
 + rewrite power_S.
 change 2 with (2*1) at 1.
 apply mult_le_compat; auto.
 apply power_ge_1; lia.
 + apply Hk1; auto.
 Qed.

 Let Hf1_1 : forall i j, i < j <= S m -> f1 i < f1 j.
 Proof.
 intros [ | i ] [ | j ] Hij; simpl; try lia.
 * apply lt_le_trans with (k 0); try lia.
 destruct j; auto; apply lt_le_weak, Hk2; lia.
 * apply Hk2; lia.
 Qed.

 Let Hf1_2 : ∑ (S (S m )) (fun i => power (f1 i ) r ) = u + power (power (S l) 2) r.
 Proof.
 rewrite msum_S; unfold f1.
 rewrite <- Hu1; auto.
 Qed.

 Let Hh_1 : k m = power (S l) 2.
 Proof.
 destruct (le_lt_dec (power (S l) 2) (k m)) as [ H | H ].
 + apply le_antisym; auto.
 + assert (∑ (S (S m )) (fun i => power (f1 i ) r ) < power (power (S l) 2) r); try lia.
 apply sum_powers_inc_lt; auto.
 - intros [ | i ] Hi; simpl.
 * apply (@power_smono_l 1 _ 2); lia.
 * apply le_lt_trans with (2 := H).
 destruct (eq_nat_dec i m); subst; auto.
 apply lt_le_weak, Hk2; lia.
 - intros; apply Hf1_1; lia.
 Qed.

 Let Hu : u = ∑ (S m ) (fun i => power (f1 i ) r ).
 Proof.
 rewrite msum_plus1 in Hf1_2; auto.
 simpl f1 at 2 in Hf1_2.
 rewrite Hh_1 in Hf1_2.
 lia.
 Qed.

 Let Huu : u *u = ∑ (S m ) (fun i => power (2*f1 i ) r )
 + ∑ (S m ) (fun i => ∑ i (fun j => 2*power (f1 i + f1 j ) r )).
 Proof.
 rewrite Hu, square_sum; f_equal.
 + apply msum_ext; intros; rewrite <- power_plus; f_equal; lia.
 + rewrite <- sum_0n_scal_l; apply msum_ext; intros i Hi.
 rewrite <- sum_0n_scal_l; apply msum_ext; intros j Hj.
 rewrite power_plus; ring.
 Qed.

 Let HSl_q : 2 * S (power (S l) 2) < power (2 * q) 2.
 Proof.
 rewrite <- (mult_2_eq_plus q), power_plus.
 apply le_lt_trans with (2*power q 2).
 + apply mult_le_compat; auto.
 apply power_smono_l; lia.
 + assert (power 1 2 < power q 2) as H.
 { apply power_smono_l; lia. }
 rewrite power_1 in H.
 apply Nat.mul_lt_mono_pos_r; lia.
 Qed.

 Let Hu1_1 : { d | d <= S m /\ u1 = ∑ d (fun i => power (2*f1 i ) r ) }.
 Proof.
 destruct const_u1_prefix with (m := S m) (k := power (S l) 2) (k' := S (power (S l) 2))
 (u := u) (w := w) (f := fun i => f1 i)
 as (d & H1 & H2); auto.
 + unfold r.
 apply le_lt_trans with (2*S (power (S l) 2)); try lia.
 apply le_lt_trans with (power (S (S (S l))) 2).
 do 4 rewrite power_S.
 * generalize (@power_ge_1 l 2); intros; lia.
 * apply power_smono_l; lia.
 + intros i Hi; generalize (@Hf1_0 i); intros; lia.
 + intros; apply Hf1_1; lia.
 + exists d; split; auto.
 rewrite H20, H2.
 apply msum_ext; intros; ring.
 Qed.

 Let Hk_final : k 0 = 4 /\ forall i, i < m -> k (S i) = 2*k i.
 Proof.
 destruct Hu1_1 as (d & Hd1 & E).
 rewrite Hu1 in E.
 apply sum_powers_injective in E; auto.
 + destruct E as (? & E); subst d; split.
 * rewrite E; try lia; auto.
 * intros; rewrite E; auto; lia.
 + intros i j H; specialize (@Hf1_1 i j); intros; lia.
 Qed.

 Let Hk_is_power i : i <= m -> k i = power (S (S i)) 2.
 Proof.
 induction i as [ | i IHi ]; intros Hi.
 + rewrite (proj1 Hk_final); auto.
 + rewrite (proj2 Hk_final), IHi, <- power_S; auto; lia.
 Qed.

 Let Hm_is_l : S m = l.
 Proof.
 rewrite Hk_is_power in Hh_1; auto.
 apply power_2_inj in Hh_1; lia.
 Qed.

 Fact obtain_u_u1_value : u = ∑ l (fun i => power (power (S i ) 2) r )
 /\ u1 = ∑ l (fun i => power (power (S (S i )) 2) r ).
 Proof using Hu.
 split.
 + rewrite <- Hm_is_l, Hu.
 apply msum_ext.
 intros [ | i ]; simpl; auto.
 intros; rewrite Hk_is_power; auto; lia.
 + rewrite <- Hm_is_l, Hu1.
 apply msum_ext.
 intros [ | i ]; simpl; auto.
 * rewrite Hk_is_power; auto; lia.
 * intros; rewrite Hk_is_power; auto; lia.
 Qed.

 End const_1_cs.

 End const_1.

 Variable (l q : nat).

 Notation r := (power (4*q) 2).

 Definition seqs_of_ones u u1 :=
 l +1 < q
 /\ u = ∑ l (fun i => power (power (S i ) 2) r )
 /\ u1 = ∑ l (fun i => power (power (S (S i )) 2) r ).

 Lemma seqs_of_ones_dio u u1 :
 seqs_of_ones u u1
 <-> l = 0 /\ u = 0 /\ u1 = 0 /\ 2 <= q
 \/ 0 < l /\ l +1 < q
 /\ exists u2 w r0 r1 p1 p2 ,
 r0 = r
 /\ r1 +1 = r0
 /\ p1 = power (1+l) 2
 /\ p2 = power p1 r0
 /\ 1+r1 *w = r0 *p2
 /\ u *u = u1 + u2
 /\ u1 = (u *u ) ⇣ w
 /\ u2 = (u *u ) ⇣ (2*w )
 /\ r0 *r0 + u1 = u + p2
 /\ divides (r0 *r0 *r0 *r0) u1.
 Proof.
 split.
 + intros (H2 & H3 & H4).
 destruct (le_lt_dec l 0) as [ H1 | H1 ].
 - assert (l =0) by lia; subst l.
 rewrite msum_0 in H3, H4; subst; left; lia.
 - right; split; auto; split; auto.
 destruct (const1_cn H1 H2 H3 H4) as (w & u2 & E1 & E2 & E3 & E4 & E5 & E6).
 exists u2, w, r, (r -1), (power (S l) 2), (power (power (S l) 2) r); repeat (split; auto).
 * generalize (@power_ge_1 (4*q) 2); intros; lia.
 * revert E5; rewrite power_S, power_1; auto.
 * revert E6; do 3 rewrite power_S; rewrite power_1.
 repeat rewrite mult_assoc; auto.
 + intros [ (H1 & H2 & H3 & H4)
 | (H1 & H2 & u2 & w & r0 & r1 & p1 & p2 & ? & H0 & ? & ? & E1 & E2 & E3 & E4 & E5 & E6) ].
 - red; subst; do 2 rewrite msum_0; lia.
 - assert (r1 = r0-1) by lia; clear H0.
 subst r0 r1 p1 p2; split; auto.
 apply obtain_u_u1_value with w u2; auto.
 * rewrite power_S, power_1; auto.
 * do 3 rewrite power_S; rewrite power_1.
 repeat rewrite mult_assoc; auto.
 Qed.

 Definition is_cipher_of f a :=
 l +1 < q
 /\ (forall i, i < l -> f i < power q 2)
 /\ a = ∑ l (fun i => f i * power (power (S i ) 2) r ).

 Fact is_cipher_of_0 f a : l = 0 -> is_cipher_of f a <-> 1 < q /\ a = 0.
 Proof.
 intros ?; unfold is_cipher_of; subst l.
 rewrite msum_0; simpl.
 repeat (split; try tauto).
 intros; lia.
 Qed.

 Fact is_cipher_of_inj f1 f2 a : is_cipher_of f1 a -> is_cipher_of f2 a -> forall i, i < l -> f1 i = f2 i.
 Proof.
 intros (H1 & H2 & H3) (_ & H4 & H5).
 rewrite H3 in H5.
 revert H5; apply power_decomp_unique.
 + apply (@power_mono_l 1 _ 2); lia.
 + intros; apply power_smono_l; lia.
 + intros i Hi; apply lt_le_trans with (1 := H2 _ Hi), power_mono_l; lia.
 + intros i Hi; apply lt_le_trans with (1 := H4 _ Hi), power_mono_l; lia.
 Qed.

 Fact is_cipher_of_fun f1 f2 a b :
 (forall i, i < l -> f1 i = f2 i )
 -> is_cipher_of f1 a
 -> is_cipher_of f2 b
 -> a = b.
 Proof.
 intros H1 (_ & _ & H2) (_ & _ & H3); subst a b.
 apply msum_ext; intros; f_equal; auto.
 Qed.

 Lemma is_cipher_of_equiv f1 f2 a b :
 is_cipher_of f1 a
 -> is_cipher_of f2 b
 -> a = b <-> forall i, i < l -> f1 i = f2 i.
 Proof.
 intros Ha Hb; split.
 + intro; subst; revert Ha Hb; apply is_cipher_of_inj.
 + intro; revert Ha Hb; apply is_cipher_of_fun; auto.
 Qed.

 Lemma is_cipher_of_const_1 u : 0 < l -> is_cipher_of (fun _ => 1) u
 <-> l +1 < q /\ exists u1 , seqs_of_ones u u1.
 Proof.
 intros Hl.
 split.
 + intros (H1 & H2 & H3); split; auto.
 exists (∑ l (fun i => power (power (S (S i )) 2) r )).
 rewrite H3; split; auto; split; auto.
 apply msum_ext; intros; ring.
 + intros (H1 & u1 & _ & H2).
 apply proj1 in H2.
 repeat (split; auto).
 * intros; apply (@power_smono_l 0); lia.
 * rewrite H2; apply msum_ext; intros; ring.
 Qed.

 Fact is_cipher_of_u : l +1 < q -> is_cipher_of (fun _ => 1) (∑ l (fun i => power (power (S i ) 2) r )).
 Proof.
 intros H; split; auto; split.
 + intros; apply (@power_mono_l 1 _ 2); lia.
 + apply msum_ext; intros; lia.
 Qed.

 Definition the_cipher f : l +1 < q -> (forall i, i < l -> f i < power q 2) -> { c | is_cipher_of f c }.
 Proof.
 intros H1 H2.
 exists (∑ l (fun i => f i * power (power (S i ) 2) r )); split; auto.
 Qed.

 Definition Code a := exists f , is_cipher_of f a.

 Lemma Code_dio a : Code a <-> l = 0 /\ 1 < q /\ a = 0
 \/ 0 < l /\ l +1 < q /\ exists p u u1 , p +1 = power q 2 /\ seqs_of_ones u u1 /\ a ≲ p *u.
 Proof.
 split.
 + intros (f & H1 & H2 & H3).
 destruct (eq_nat_dec l 0) as [ Hl | Hl ].
 * left; subst l; rewrite msum_0 in H3; lia.
 * right; split; try lia; split; auto.
 exists (power q 2-1), (∑ l (fun i => power (power (S i ) 2) r )), (∑ l (fun i => power (power (S (S i )) 2) r )).
 repeat (split; auto).
 - generalize (@power_ge_1 q 2); intros; lia.
 - rewrite H3.
 apply sum_power_binary_lt with (q := 4*q); auto; try lia.
 intros; apply power_smono_l; lia.
 + intros [ (H1 & H2 & H3) | (H1 & H2 & p & u1 & u2 & ? & H3 & H4) ].
 * exists (fun _ => 0); subst a; apply is_cipher_of_0; auto.
 * destruct H3 as (_ & H3 & _).
 assert (p = power q 2 -1) by lia; subst p.
 rewrite H3 in H4.
 apply sum_power_binary_lt_inv with (q := 4*q) (e := fun i => power (S i) 2) in H4; auto; try lia.
 2,3: intros; apply power_smono_l; lia.
 destruct H4 as (f & H4 & H5).
 exists f; split; auto.
 Qed.

 Definition Const c v := exists f , is_cipher_of f v /\ forall i, i < l -> f i = c.

 Lemma Const_dio c v : Const c v <-> l = 0 /\ 1 < q /\ v = 0
 \/ 0 < l /\ l +1 < q /\
 exists p u u1 , p = power q 2 /\ c < p /\ seqs_of_ones u u1 /\ v = c *u.
 Proof.
 split.
 + intros (f & (H1 & H2 & H3) & H4).
 destruct (eq_nat_dec l 0) as [ Hl | Hl ].
 * left; subst l; rewrite msum_0 in H3; lia.
 * right; split; try lia; split; auto.
 exists (power q 2), (∑ l (fun i => power (power (S i ) 2) r )), (∑ l (fun i => power (power (S (S i )) 2) r )).
 repeat (split; auto).
 - rewrite <- (H4 0); try lia; apply H2; lia.
 - rewrite H3, <- sum_0n_scal_l; apply msum_ext.
 intros; f_equal; auto.
 + intros [ (H1 & H2 & H3) | (H1 & H2 & p & u1 & u2 & ? & H3 & H4 & H5) ].
 * exists (fun _ => 0); subst v; split.
 - apply is_cipher_of_0; auto.
 - subst l; intros; lia.
 * destruct H4 as (_ & H4 & _).
 rewrite H4, <- sum_0n_scal_l in H5.
 exists (fun _ => c); split; auto.
 split; auto; split; auto.
 intros; lia.
 Qed.

 Let Hr : 1 < q -> 4 <= r.
 Proof.
 intros H.
 replace (4*q) with (2*q +2*q) by lia.
 rewrite power_plus.
 change 4 with ((power 1 2)*(power 1 2)); apply mult_le_compat;
 apply power_mono_l; try lia.
 Qed.

 Section plus.

 Variable (a b c : nat -> nat) (ca cb cc : nat)
 (Ha : is_cipher_of a ca)
 (Hb : is_cipher_of b cb)
 (Hc : is_cipher_of c cc).

 Definition Code_plus := ca = cb + cc.

 Lemma Code_plus_spec : Code_plus <-> forall i, i < l -> a i = b i + c i.
 Proof using Ha Hb Hc.
 symmetry; unfold Code_plus.
 destruct Ha as (H & Ha1 & Ha2).
 destruct Hb as (_ & Hb1 & Hb2).
 destruct Hc as (_ & Hc1 & Hc2).
 destruct (eq_nat_dec l 0) as [ | Hl ].
 + clear Hc Hb Ha. subst l; rewrite msum_0 in *; split; intros; lia.
 + rewrite Hc2, Ha2, Hb2, <- sum_0n_distr_in_out.
 split.
 * intros; apply msum_ext; intros; f_equal; auto.
 * intros E i Hi.
 apply power_decomp_unique with (i := i) in E; auto; try lia; clear i Hi.
 - intros; apply power_smono_l; lia.
 - intros i Hi; apply lt_le_trans with (1 := Ha1 _ Hi), power_mono_l; lia.
 - intros i Hi.
 apply lt_le_trans with (power (S q) 2).
 ++ rewrite power_S, <- mult_2_eq_plus.
 generalize (Hb1 _ Hi) (Hc1 _ Hi); lia.
 ++ apply power_mono_l; lia.
 Qed.

 End plus.

 Notation u := (∑ l (fun i => power (power (S i ) 2) r )).
 Notation u1 := (∑ l (fun i => power (power (S (S i )) 2) r )).

 Section mult_utils.

 Variable (b c : nat -> nat) (cb cc : nat)
 (Hb : is_cipher_of b cb)
 (Hc : is_cipher_of c cc).

 Let eq1 : cb *cc = ∑ l (fun i => (b i *c i )*power (power (S (S i )) 2) r )
 + ∑ l (fun i => ∑ i (fun j => (b i *c j + b j *c i )*power (power (S i ) 2 + power (S j ) 2) r )).
 Proof.
 destruct Hb as (H1 & Hb1 & Hb2).
 destruct Hc as (_ & Hc1 & Hc2).
 rewrite Hb2, Hc2, product_sums; f_equal.
 * apply msum_ext; intros; rewrite (power_S (S _)).
 rewrite <- (mult_2_eq_plus (power _ _)), power_plus; ring.
 * apply msum_ext; intros i Hi.
 apply msum_ext; intros j Hj.
 rewrite power_plus; ring.
 Qed.

 Let Hbc_1 i : i < l -> b i * c i < r.
 Proof.
 destruct Hb as (H1 & Hb1 & Hb2).
 destruct Hc as (_ & Hc1 & Hc2).
 intro; apply mult_lt_power_2_4; auto.
 Qed.

 Let Hbc_2 i j : i < l -> j < l -> b i * c j + b j * c i < r.
 Proof.
 destruct Hb as (H1 & Hb1 & Hb2).
 destruct Hc as (_ & Hc1 & Hc2).
 intros; apply mult_lt_power_2_4'; auto.
 Qed.

 Let Hbc_3 : ∑ l (fun i => (b i *c i )*power (power (S (S i )) 2) r )
 = msum nat_join 0 l (fun i => (b i *c i )*power (power (S (S i)) 2) r).
 Proof.
 destruct Hb as (H1 & Hb1 & Hb2).
 destruct Hc as (_ & Hc1 & Hc2).
 apply sum_powers_ortho with (q := 4*q); try lia; auto.
 intros ? ? ? ? E; apply power_2_inj in E; lia.
 Qed.

 Let Hbc_4 : ∑ l (fun i => ∑ i (fun j => (b i *c j + b j *c i )*power (power (S i ) 2 + power (S j ) 2) r ))
 = msum nat_join 0 l (fun i =>
 msum nat_join 0 i (fun j => (b i *c j + b j *c i )*power (power (S i) 2 + power (S j) 2) r)).
 Proof.
 destruct Hb as (H1 & Hb1 & Hb2).
 destruct Hc as (_ & Hc1 & Hc2).
 rewrite double_sum_powers_ortho with (q := 4*q); auto; try lia.
 + intros; apply Hbc_2; lia.
 + intros ? ? ? ? ? ? E; apply sum_2_power_2_injective in E; lia.
 Qed.

 Let eq2 : cb *cc = msum nat_join 0 l (fun i => (b i *c i )*power (power (S (S i)) 2) r)
 ⇡ msum nat_join 0 l (fun i =>
 msum nat_join 0 i (fun j => (b i *c j + b j *c i )*power (power (S i) 2 + power (S j) 2) r)).
 Proof.
 destruct Hb as (H1 & Hb1 & Hb2).
 rewrite eq1, Hbc_3, Hbc_4.
 apply nat_ortho_plus_join.
 apply nat_ortho_joins.
 intros i j Hi Hj; apply nat_ortho_joins_left.
 intros k Hk.
 apply nat_meet_powers_neq with (q := 4*q); auto; try lia.
 * apply power_2_n_ij_neq; lia.
 * apply Hbc_2; lia.
 Qed.

 Let Hr_1 : (r -1)*u1 = ∑ l (fun i => (r -1)*power (power (S (S i )) 2) r ).
 Proof. rewrite sum_0n_scal_l; auto. Qed.

 Let Hr_2 : (r -1)*u1 = msum nat_join 0 l (fun i => (r -1)*power (power (S (S i)) 2) r).
 Proof.
 destruct Hb as (H1 & Hb1 & Hb2).
 destruct Hc as (_ & Hc1 & Hc2).
 rewrite Hr_1.
 apply sum_powers_ortho with (q := 4*q); auto; try lia.
 intros ? ? ? ? E; apply power_2_inj in E; lia.
 Qed.

 Fact cipher_mult_eq : (cb *cc )⇣((r -1)*u1 ) = ∑ l (fun i => (b i *c i )*power (power (S (S i )) 2) r ).
 Proof using Hb Hc.
 destruct Hb as (H1 & Hb1 & Hb2).
 destruct Hc as (_ & Hc1 & Hc2).
 rewrite eq2, Hbc_3, Hr_2.
 rewrite nat_meet_comm, nat_meet_join_distr_l.
 rewrite <- Hr_2 at 1; rewrite Hr_1, <- Hbc_3.
 rewrite meet_sum_powers with (q := 4*q); auto; try (intros; lia).
 2: intros; apply power_smono_l; lia.
 rewrite (proj2 (nat_ortho_joins _ _ _ _)), nat_join_n0.
 * apply msum_ext; intros i Hi; f_equal.
 rewrite nat_meet_comm; apply binary_le_nat_meet, power_2_minus_1_gt; auto.
 * intros i j Hi Hj.
 apply nat_ortho_joins_left.
 intros k Hk; apply nat_meet_powers_neq with (q := 4*q); auto; try lia.
 + apply power_2_n_ij_neq; lia.
 + apply Hbc_2; lia.
 Qed.

 End mult_utils.

 Section mult.

 Variable (a b c : nat -> nat) (ca cb cc : nat)
 (Ha : is_cipher_of a ca)
 (Hb : is_cipher_of b cb)
 (Hc : is_cipher_of c cc).

 Definition Code_mult :=
 l = 0
 \/ l <> 0
 /\ exists v v1 r' r'' p ,
 r'' = r
 /\ r'' = r'+1
 /\ seqs_of_ones v v1
 /\ p = (ca *v )⇣(r'*v1 )
 /\ p = (cb *cc )⇣(r'*v1 ).

 Lemma Code_mult_spec : Code_mult <-> forall i, i < l -> a i = b i * c i.
 Proof using Ha Hb Hc.
 unfold Code_mult; symmetry.
 destruct Ha as (Hlq & Ha1 & Ha2).
 destruct Hb as (_ & Hb1 & Hb2).
 destruct Hc as (_ & Hc1 & Hc2).
 destruct (eq_nat_dec l 0) as [ | Hl ].
 + rewrite e in *; split; intros; auto; lia.
 + split.
 * intros H; right; split; try lia.
 exists u, u1, (r -1), r, (ca * u ⇣ ((r -1) * u1 )).
 split; auto; split; try lia.
 repeat (split; auto).
 generalize (is_cipher_of_u Hlq); intros H2.
 rewrite cipher_mult_eq with (1 := Ha) (2 := H2).
 rewrite cipher_mult_eq with (1 := Hb) (2 := Hc).
 apply msum_ext; intros; rewrite H; try ring; lia.
 * intros [ | (_ & v & v1 & r' & r'' & p & H0 & H1 & H2 & H3 & H4) ]; try (destruct Hl; auto; fail).
 destruct H2 as (_ & ? & ?); subst v v1.
 rewrite H3 in H4.
 revert H4.
 generalize (is_cipher_of_u Hlq); intros H2.
 replace r' with (r -1) by lia.
 rewrite cipher_mult_eq with (1 := Ha) (2 := H2).
 rewrite cipher_mult_eq with (1 := Hb) (2 := Hc).
 intros E.
 intros i Hi.
 rewrite <- power_decomp_unique with (5 := E); auto; try lia.
 - intros; apply power_smono_l; lia.
 - intros j Hj; rewrite Nat.mul_1_r.
 apply lt_le_trans with (1 := Ha1 _ Hj), power_mono_l; lia.
 - intros; apply mult_lt_power_2_4; auto.
 Qed.

 End mult.

 Section inc_seq.

 Definition CodeNat c := is_cipher_of (fun i => i) c.

 Local Lemma IncSeq_dio_priv y : CodeNat y <-> l = 0 /\ 1 < q /\ y = 0
 \/ 0 < l
 /\ exists z v v1 ,
 seqs_of_ones v v1
 /\ Code y
 /\ Code z
 /\ y + l *(power (power (S l) 2) r ) = (z *v )⇣((r -1) * v1 )
 /\ y +v1 +power (power 1 2) r = z + power (power (S l) 2) r.
 Proof.
 split.
 + intros (H1 & H2 & H3).
 destruct (le_lt_dec l 0) as [ | Hl ].
 - assert (l = 0) as -> by lia.
 rewrite msum_0 in H3; left; lia.
 - right; split; auto.
 exists (∑ l (fun i => (S i ) * power (power (S i ) 2) r )), u, u1; split; auto.
 { split; auto. }
 split.
 { rewrite H3; exists (fun i => i); split; auto. }
 split.
 { exists S; repeat (split; auto).
 intros; apply lt_le_trans with q; try lia.
 apply power_ge_n; auto. }
 split.
 { rewrite cipher_mult_eq with (b := S) (c := fun _ => 1).
 * rewrite H3.
 rewrite <- msum_plus1 with (f := fun i => i *power (power (S i) 2) r); auto.
 rewrite msum_S, Nat.mul_0_l, Nat.add_0_l.
 apply msum_ext; intros; ring.
 * repeat split; auto; intros.
 apply lt_le_trans with q; try lia.
 apply power_ge_n; auto.
 * apply is_cipher_of_u; auto. }
 { rewrite H3.
 destruct l as [ | l' ]; try lia.
 rewrite msum_S, Nat.mul_0_l, Nat.add_0_l.
 rewrite msum_plus1; auto.
 rewrite plus_assoc.
 rewrite msum_S.
 rewrite <- msum_sum; auto.
 2: intros; ring.
 rewrite Nat.mul_1_l, plus_comm.
 repeat rewrite <- plus_assoc; do 2 f_equal.
 apply msum_ext; intros; ring. }
 + intros [ (H1 & H2 & H3) | (Hl & z & v & v1 & H1 & H2 & H3 & H4 & H5) ].
 - split; subst; auto; split; intros; try lia.
 rewrite msum_0; auto.
 - destruct H1 as (Hq & ? & ?); subst v v1.
 split; auto; split.
 { intros i Hi; apply lt_le_trans with q; try lia.
 apply power_ge_n; auto. }
 destruct H2 as (f & Hf).
 destruct H3 as (g & Hg).
 generalize (is_cipher_of_u Hq); intros Hu.
 rewrite cipher_mult_eq with (1 := Hg) (2 := Hu) in H4.
 destruct Hf as (_ & Hf & Hy).
 destruct Hg as (_ & Hg & Hz).
 set (h i := if le_lt_dec l i then l else f i).
 assert (y+l *power (power (S l) 2) r = ∑ (S l ) (fun i => h i * power (power (S i ) 2) r )) as H6.
 { rewrite msum_plus1; auto; f_equal.
 * rewrite Hy; apply msum_ext.
 intros i Hi; unfold h.
 destruct (le_lt_dec l i); try lia.
 * unfold h.
 destruct (le_lt_dec l l); try lia. }
 rewrite H4 in H6.
 set (g' i := match i with 0 => 0 | S i => g i end).
 assert ( ∑ (S l ) (fun i => g' i * power (power (S i ) 2) r )
 = ∑ l (fun i : nat => g i * 1 * power (power (S (S i )) 2) r )) as H7.
 { unfold g'; rewrite msum_S; apply msum_ext; intros; ring. }
 rewrite <- H7 in H6.
 assert (forall i, i < S l -> g' i = h i) as H8.
 { apply power_decomp_unique with (5 := H6); try lia.
 * intros; apply power_smono_l; lia.
 * unfold g'; intros [ | i ] Hi; try lia.
 apply lt_S_n in Hi.
 apply lt_le_trans with (1 := Hg _ Hi), power_mono_l; lia.
 * intros i Hi; unfold h.
 destruct (le_lt_dec l i) as [ | Hi' ].
 + apply lt_le_trans with (4*q); try lia.
 apply power_ge_n; auto.
 + apply lt_le_trans with (1 := Hf _ Hi'), power_mono_l; lia. }
 assert (h 0 = 0) as E0.
 { rewrite <- H8; simpl; lia. }
 assert (forall i, i < l -> h (S i) = g i) as E1.
 { intros i Hi; rewrite <- H8; simpl; lia. }
 assert (f 0 = 0) as E3.
 { unfold h in E0; destruct (le_lt_dec l 0); auto; lia. }
 assert (forall i, S i < l -> f (S i) = g i) as E4.
 { intros i Hi; specialize (E1 i); unfold h in E1.
 destruct (le_lt_dec l (S i)); lia. }
 assert (g (l -1) = l) as E5.
 { specialize (E1 (l -1)); unfold h in E1.
 destruct (le_lt_dec l (S (l -1))); lia. }
 clear H6 H7 g' H8 E0 E1 h H4.
 assert (y + u1 + power (power 1 2) r =
 ∑ l (fun i => (1+f i ) * power (power (S i ) 2) r )
 + power (power (S l) 2) r) as E1.
 { rewrite sum_0n_distr_in_out.
 rewrite <- Hy, sum_0n_scal_l, Nat.mul_1_l.
 destruct l as [ | l' ]; try lia.
 rewrite msum_plus1; auto.
 rewrite msum_S; ring. }
 assert (forall i, i < l -> 1+f i = g i) as E2.
 { apply power_decomp_unique with (f := fun i => power (S i) 2) (p := r); try lia.
 + intros; apply power_smono_l; lia.
 + intros i Hi; apply le_lt_trans with (power q 2); auto.
 * apply Hf; auto.
 * apply power_smono_l; lia.
 + intros i Hi; apply lt_le_trans with (1 := Hg _ Hi), power_mono_l; lia. }
 rewrite Hy; apply msum_ext.
 clear Hy Hf Hg Hz H5 E5 E1 Hu.
 intros i Hi; f_equal; revert i Hi.
 induction i as [ | i IHi ]; intros Hi; auto.
 rewrite E4, <- E2; try lia.
 Qed.

 Lemma CodeNat_dio y : CodeNat y <-> l = 0 /\ 1 < q /\ y = 0
 \/ 0 < l
 /\ exists z v v1 p0 p1 p2 r1 ,
 p0 = r
 /\ r1 +1 = p0
 /\ p1 = power (1+l) 2
 /\ p2 = power p1 p0
 /\ seqs_of_ones v v1
 /\ Code y
 /\ Code z
 /\ y + l *p2 = (z *v ) ⇣ (r1 * v1 )
 /\ y + v1 + p0 *p0 = z + p2.
 Proof.
 rewrite IncSeq_dio_priv; split; (intros [ H | H ]; [ left | right ]); auto; revert H;
 intros (H1 & H); split; auto; clear H1; revert H.
 + intros (z & v & v1 & H1 & H2 & H3 & H4 & H5).
 exists z, v, v1, r, (power (S l) 2), (power (power (S l) 2) r), (r -1); repeat (split; auto).
 * destruct H1; lia.
 * rewrite <- H5; f_equal.
 rewrite power_1, power_S, power_1; auto.
 + intros (z & v & v1 & p0 & p1 & p2 & r1 & H1 & H2 & H3 & H4 & H5 & H6 & H7 & H8 & H9).
 assert (r1 = r - 1) by lia; clear H2; subst.
 exists z, v, v1; repeat (split; auto).
 simpl in H9 |- *; rewrite <- H9; f_equal.
 rewrite power_1, power_S, power_1; auto.
 Qed.

 End inc_seq.

End sums.