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import Mathlib
import Mathlib.Analysis.SpecialFunctions.Pow.Real
set_option linter.unusedVariables.analyzeTactics true
open Real Set
lemma mylemma_1
(x a: ℕ → ℝ)
(hxp: ∀ (i : ℕ), 0 < x i)
(h₀: ∀ (n : ℕ),
1 ≤ n ∧ n ≤ 2023 →
a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
* Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) :
∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2022) → a (n) < a (n + 1) := by
intros n hn
have h₂: a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
* Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
refine h₀ n ?_
constructor
. exact hn.1
linarith
have h₃: a (n + 1) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k)
* Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k) := by
refine h₀ (n + 1) ?_
constructor
. linarith
linarith
rw [h₂,h₃]
refine sqrt_lt_sqrt ?_ ?_
. refine le_of_lt ?_
refine mul_pos ?_ ?_
. refine Finset.sum_pos ?_ ?_
. exact fun i _ => hxp i
. simp
linarith
. refine Finset.sum_pos ?_ ?_
intros i _
exact one_div_pos.mpr (hxp i)
. simp
linarith
have g₀: 1 ≤ n + 1 := by linarith
rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _]
repeat rw [add_mul, mul_add]
have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) +
x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by
refine add_pos ?_ ?_
. refine mul_pos ?_ ?_
. refine Finset.sum_pos ?_ ?_
. exact fun i _ => hxp i
. simp
linarith
. exact one_div_pos.mpr (hxp (n + 1))
. refine mul_pos ?_ ?_
. exact hxp (n + 1)
. refine add_pos ?_ ?_
. refine Finset.sum_pos ?_ ?_
. intros i _
exact one_div_pos.mpr (hxp i)
. simp
linarith
exact one_div_pos.mpr (hxp (n + 1))
linarith
lemma mylemma_amgm
(b1 b2 b3 b4 :ℝ)
(hb1: 0 ≤ b1)
(hb2: 0 ≤ b2)
(hb3: 0 ≤ b3)
(hb4: 0 ≤ b4) :
(4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4) := by
let w1 : ℝ := (4:ℝ)⁻¹
let w2 : ℝ := w1
let w3 : ℝ := w2
let w4 : ℝ := w3
rw [mul_comm]
refine mul_le_of_le_div₀ ?_ (by norm_num) ?_
. refine add_nonneg ?_ hb4
refine add_nonneg ?_ hb3
exact add_nonneg hb1 hb2
have h₀: (b1^w1 * b2^w2 * b3^w3 * b4^w4) ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4 := by
refine geom_mean_le_arith_mean4_weighted (by norm_num) ?_ ?_ ?_ hb1 hb2 hb3 hb4 ?_
. norm_num
. norm_num
. norm_num
. norm_num
repeat rw [mul_rpow _]
ring_nf at *
linarith
repeat { assumption }
. exact mul_nonneg hb1 hb2
. exact hb4
. refine mul_nonneg ?_ hb3
exact mul_nonneg hb1 hb2
lemma mylemma_2
(x a: ℕ → ℝ)
(hxp: ∀ (i : ℕ), 0 < x i)
(h₀: ∀ (n : ℕ),
1 ≤ n ∧ n ≤ 2023 →
a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
* (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)))
(n: ℕ)
(hn: 1 ≤ n ∧ n ≤ 2021) :
(4 * a n ≤
(Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) +
(x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
repeat rw [mul_add, add_mul]
have g₁₁: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1 := by
refine le_of_lt ?_
refine Finset.sum_pos ?_ ?_
. exact fun i _ => hxp i
. simp
linarith
have g₁₂: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹ := by
refine le_of_lt ?_
refine Finset.sum_pos ?_ ?_
. intros i _
exact inv_pos.mpr (hxp i)
. simp
linarith
have h₃₂: 4 * ( ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) *
((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) *
((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) *
(x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹
≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) +
(Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) +
((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) +
x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
let b1:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))
let b2:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))
let b3:ℝ := (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)
let b4:ℝ := x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)
have hb1: b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by
exact rfl
have hb2: b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) := by
exact rfl
have hb3: b3 = (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
exact rfl
have hb4: b4 = x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
exact rfl
rw [← hb1, ← hb2, ← hb3, ← hb4]
have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by
have b1p: 0 ≤ b1 := by
rw [hb1]
refine mul_nonneg ?_ ?_
. ring_nf
exact g₁₁
. refine le_of_lt ?_
exact one_div_pos.mpr (hxp (n + 1))
have b2p: 0 ≤ b2 := by
rw [hb2]
refine mul_nonneg ?_ ?_
. ring_nf
exact g₁₁
. refine le_of_lt ?_
exact one_div_pos.mpr (hxp (n + 2))
have b3p: 0 ≤ b3 := by
rw [hb3]
refine mul_nonneg ?_ ?_
. exact LT.lt.le (hxp (n + 1))
. ring_nf
exact g₁₂
have b4p: 0 ≤ b4 := by
rw [hb4]
refine mul_nonneg ?_ ?_
. exact LT.lt.le (hxp (n + 2))
. ring_nf
exact g₁₂
exact mylemma_amgm b1 b2 b3 b4 b1p b2p b3p b4p
linarith
have h₃₃: 4 * a (n) = 4 * (((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) *
((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) *
((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) *
(x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ := by
simp
ring_nf
have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2
* x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ *
(Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2
= x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ *
(Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 *
(Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by
linarith
have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by
rw [mul_assoc]
have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by
refine div_self ?_
exact ne_of_gt (hxp (1 + n))
have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by
refine div_self ?_
exact ne_of_gt (hxp (2 + n))
rw [gg₁, gg₂]
norm_num
rw [g₁] at g₀
rw [g₀]
simp
repeat rw [mul_rpow]
have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹
= (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by
rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹]
norm_num
have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹
= (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by
rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹]
norm_num
-- rw [g₂, ← sqrt_eq_rpow (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1)]
-- rw [g₃, ← sqrt_eq_rpow (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹)]
have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
* Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
refine h₀ n ?_
constructor
. exact hn.1
. linarith
norm_cast at *
rw [g₂, g₃, ← mul_rpow]
rw [← sqrt_eq_rpow]
ring_nf at g₄
exact g₄
. exact g₁₁
. exact g₁₂
. exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1)
. exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹)
exact Eq.trans_le h₃₃ h₃₂
lemma mylemma_3
(x a: ℕ → ℝ)
(hxp: ∀ (i : ℕ), 0 < x i)
(hx: ∀ (i j : ℕ), i ≠ j → x i ≠ x j)
(h₀: ∀ (n : ℕ),
1 ≤ n ∧ n ≤ 2023 →
a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
* (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)))
(h₀₁: ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) :
(∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) := by
intros n hn
have g₀: 0 ≤ a n + 2 := by
refine le_of_lt ?_
refine add_pos ?_ (by norm_num)
refine h₀₁ n ?_
constructor
. exact hn.1
. linarith
have g₁: 0 ≤ a (n + 2) := by
refine le_of_lt ?_
refine h₀₁ (n + 2) ?_
constructor
. linarith
. linarith
rw [← sqrt_sq g₀, ← sqrt_sq g₁]
have g₂: 0 ≤ (a n + 2) ^ 2 := by exact sq_nonneg (a n + 2)
-- simp
refine Real.sqrt_lt_sqrt g₂ ?_
have g₃: 0 ≤ ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
* (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by
refine le_of_lt ?_
refine mul_pos ?_ ?_
. refine Finset.sum_pos ?_ ?_
. exact fun i _ => hxp i
. simp
linarith
. refine Finset.sum_pos ?_ ?_
. intros i _
exact one_div_pos.mpr (hxp i)
. simp
linarith
have gn₀: a (n) ^ 2 = ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
* (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by
rw [← sq_sqrt g₃]
have g₄: 0 ≤ a n := by
refine le_of_lt ?_
refine h₀₁ n ?_
constructor
. exact hn.1
. linarith
refine (sq_eq_sq₀ g₄ ?_).mpr ?_
. exact
sqrt_nonneg
((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) *
Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)
. refine h₀ (n) ?_
constructor
. exact hn.1
. linarith
have gn₁: a (n + 2) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k)
* Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k) := by
refine h₀ (n + 2) ?_
constructor
. linarith
. linarith
rw [add_sq, gn₁, sq_sqrt]
. have ga₀: 1 ≤ n + 2 := by linarith
rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _]
have ga₁: 1 ≤ n + 1 := by linarith
rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _]
rw [add_assoc, add_assoc, add_assoc]
rw [add_mul, mul_add]
rw [← gn₀]
repeat rw [add_assoc]
refine add_lt_add_left ?_ (a (n) ^ 2)
rw [mul_add (x (n + 1) + x (n + 2))]
have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by
repeat rw [add_mul, mul_add, mul_add]
repeat rw [mul_div_left_comm _ 1 _, one_mul]
repeat rw [div_self ?_]
. have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2))
= x (n + 1) * x (n + 1) := by
rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_]
simp
exact ne_of_gt (hxp (n + 2))
have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1))
= x (n + 2) * x (n + 2) := by
rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_]
simp
exact ne_of_gt (hxp (n + 1))
have h₂₀: 0 < x (n + 1) * x (n + 2) := by
refine mul_pos ?_ ?_
. exact hxp (n + 1)
. exact hxp (n + 2)
have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by
refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀)
rw [mul_add, hc₂, hc₃, ← sq, ← sq]
refine lt_of_sub_pos ?_
have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2
= (x (n + 1) - x (n + 2)) ^ 2 := by
rw [sub_sq]
linarith
rw [gh₂₁]
refine (sq_pos_iff).mpr ?_
refine sub_ne_zero.mpr ?_
exact hx (n+1) (n+2) (by linarith)
linarith
. exact ne_of_gt (hxp (n + 2))
. exact ne_of_gt (hxp (n + 1))
clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁
have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
* (1 / x (n + 1) + 1 / x (n + 2)) +
((x (n + 1) + x (n + 2))
* Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
exact mylemma_2 (fun k => x k) a hxp h₀ n hn
linarith
. refine mul_nonneg ?_ ?_
. refine Finset.sum_nonneg ?_
intros i _
exact LT.lt.le (hxp i)
. refine Finset.sum_nonneg ?_
intros i _
simp
exact LT.lt.le (hxp i)
theorem imo_2023_p4
(x : ℕ → ℝ)
(a : ℕ → ℝ)
(hxp: ∀ (i: ℕ), (0 < x i) )
(hx: ∀ (i j: ℕ), (i ≠ j) → (x i ≠ x j) )
(h₀: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2023) →
a n = Real.sqrt ( (Finset.sum (Finset.Ico 1 (n + 1)) fun (k : ℕ) => (x k))
* (Finset.sum (Finset.Ico 1 (n + 1)) fun (k : ℕ) => 1 / (x k)) ) )
(h₁: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2023) → ∃ (kz:ℤ), (a n = ↑kz )) :
(3034 ≤ a 2023) := by
have ha1: a 1 = 1 := by
have g₀: sqrt ((Finset.sum (Finset.Ico 1 (1 + 1)) fun k => x k)
* Finset.sum (Finset.Ico 1 (1 + 1)) fun k => 1 / x k) = 1 := by
norm_num
refine div_self ?_
exact ne_of_gt (hxp 1)
rw [← g₀]
exact h₀ (1) (by norm_num)
have h₀₁: ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2023) → 0 < a n := by
intros n hn
have ha: a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k)
* Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by
exact h₀ (n) (hn)
rw [ha]
refine Real.sqrt_pos.mpr ?_
refine mul_pos ?_ ?_
. refine Finset.sum_pos ?_ ?_
. intros i _
exact hxp i
simp
linarith
. refine Finset.sum_pos ?_ ?_
. intros i _
exact one_div_pos.mpr (hxp i)
simp
linarith
have h₁₁: ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2023) → ∃ (kn:ℕ), a n = ↑kn := by
intros n hn
have g₁₁: 0 < a n := by
exact h₀₁ n hn
let ⟨p, gp⟩ := h₁ n hn
let q:ℕ := Int.toNat p
have g₁₂: ↑q = p := by
refine Int.toNat_of_nonneg ?_
rw [gp] at g₁₁
norm_cast at g₁₁
exact Int.le_of_lt g₁₁
use q
rw [gp]
norm_cast
exact id g₁₂.symm
have h₂₁: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 2 < a (n+2) := by
exact fun n a_1 => mylemma_3 (fun i => x i) a hxp hx h₀ h₀₁ n a_1
have h₂: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 3 ≤ a (n+2) := by
intros n hn
have g₀: a n + 2 < a (n + 2) := by exact h₂₁ n hn
have g₀₁: ∃ (p:ℕ), a n = ↑p := by
apply h₁₁
constructor
. linarith
. linarith
have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by
apply h₁₁
constructor
. linarith
. linarith
let ⟨p, _⟩ := g₀₁
let ⟨q, _⟩ := g₀₂
have g₁: p + 2 < q := by
suffices g₁₁: ↑p + (2:ℝ) < ↑q
. norm_cast at g₁₁
. linarith
have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast
linarith
have h₃: ∀ (n:ℕ), (0 ≤ n ∧ n ≤ 1010) → a 1 + 3 * (↑(n) + 1) ≤ a (3 + 2 * n) := by
intros n hn
induction' n with d hd
· simp
exact h₂ (1) (by norm_num)
· rw [mul_add]
simp
have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by
refine h₂ (3 + 2 * d) ?_
constructor
. linarith
. linarith
have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by
refine add_le_add_right ?_ (3)
apply hd
constructor
. linarith
. linarith
refine le_trans (by linarith[g₁]) g₀
rw [ha1] at h₃
have h₄: (3034:ℝ) = 1 + 3 * (↑1010 + 1) := by norm_num
rw [h₄]
exact h₃ (1010) (by norm_num)
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