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import Mathlib
set_option linter.unusedVariables.analyzeTactics true
open NNReal Nat BigOperators Finset
-- imo-official.org/problems/IMO2007SL.pdf
lemma aux1
(a : ℕ → NNReal)
(m : ℕ)
(hm₀ : Nat.succ 4 ≤ m) :
a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2 ≤ ∑ x ∈ Finset.range m, a (x + 1) ^ 2 := by
let fs: Finset ℕ := {0, 1, m-2, m-1}
have h₀: fs = {0, 1, m-2, m-1} := by rfl
have h₁: fs ⊆ Finset.range m := by
refine insert_subset ?_ ?_
. refine mem_range.mpr ?_
exact zero_lt_of_lt hm₀
. refine insert_subset ?_ ?_
. refine mem_range.mpr ?_
linarith
. refine insert_subset ?_ ?_
. refine mem_range.mpr ?_
refine sub_lt ?_ (by norm_num)
exact zero_lt_of_lt hm₀
. refine singleton_subset_iff.mpr ?_
refine mem_range.mpr ?_
exact sub_one_lt_of_lt hm₀
rw [← Finset.sum_sdiff h₁]
have h₂: ∑ x ∈ fs, a (x + 1) ^ 2 = a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2 := by
rw [h₀]
have g₀: 0 ∈ fs := by exact mem_insert_self 0 {1, m - 2, m - 1}
rw [← Finset.add_sum_erase fs _ g₀]
simp
have g₁: 4 ≤ m - 1 := by exact Nat.le_sub_one_of_lt hm₀
have g₂: 3 ≤ m - 2 := by exact le_sub_of_add_le hm₀
have g₃: fs.erase 0 = ({1, m - 2, m - 1}:(Finset ℕ)) := by
rw [h₀]
refine erase_insert ?h
refine forall_mem_not_eq'.mp ?_
intros b hb₀ hb₁
rw [hb₁] at hb₀
norm_num at hb₀
cases' hb₀ with hb₀ hb₀
. rw [← hb₀] at g₂
linarith
. rw [← hb₀] at g₁
linarith
rw [g₃]
have g₄: (1:ℕ) ∈ ({1, m - 2, m - 1}:(Finset ℕ)) := by
exact mem_insert_self 1 {m - 2, m - 1}
rw [← Finset.add_sum_erase _ _ g₄]
simp
rw [Finset.erase_eq_self.mpr ?_]
. have g₅: (m - 2) ∈ ({m - 2, m - 1}:(Finset ℕ)) := by
exact mem_insert_self (m - 2) {m - 1}
rw [← Finset.add_sum_erase _ _ g₅]
simp
rw [Finset.erase_eq_self.mpr ?_]
. rw [Finset.sum_singleton, Nat.sub_add_cancel (by linarith)]
rw [← Nat.sub_add_comm (by linarith)]
simp
ring_nf
. refine Finset.not_mem_singleton.mpr ?_
omega
. refine forall_mem_not_eq'.mp ?_
intros b hb₀ hb₁
rw [hb₁] at hb₀
simp at hb₀
cases' hb₀ with hb₀ hb₀
. rw [← hb₀] at g₂
linarith
. rw [← hb₀] at g₁
linarith
rw [add_comm _ (∑ x ∈ fs, a (x + 1) ^ 2), h₂]
exact le_self_add
lemma aux2
(a : ℕ → NNReal) :
∀ (n : ℕ),
4 < n ∧ n < 101 →
(∀ (x y : ℕ), x % n = y % n → a (x + 1) = a (y + 1)) →
∑ x ∈ range n, (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) ≤
(∑ x ∈ range n, a (x + 1) ^ 2) ^ 2 := by
intro n hn₀ hn₂
cases' hn₀ with hn₀ hn₁
have hn₃: n = (n - 2) + 1 + 1 := by omega
nth_rw 1 [hn₃,]
rw [Finset.sum_range_succ, sum_range_succ]
have hn₄: a (n - 2 + 1) = a (n - 1) := by
refine congrArg a (by omega)
have hn₅: a (n - 2 + 3) = a 1 := by
refine hn₂ (n - 2 + 2) 0 ?_
rw [Nat.zero_mod, Nat.sub_add_cancel ?_]
. rw [Nat.mod_self n]
. linarith
have hn₆: a (n - 2 + 1 + 3) = a 2 := by
refine hn₂ (n - 2 + 3) 1 ?_
symm
rw [Nat.mod_eq_of_lt (by linarith)]
have g₀: n - 2 + 3 = n + 1 := by linarith
rw [g₀]
refine Eq.symm (mod_eq_of_modEq ?_ (by linarith))
exact Nat.add_modEq_left
rw [← hn₃, hn₄, hn₅, hn₆]
refine le_induction ?_ ?_ n hn₀
. repeat rw [Finset.sum_range_succ]
simp
ring_nf
repeat refine add_le_add_right ?_ _
refine le_of_eq ?_
rfl
. intros m hm₀ hm₁
have hm₂: m + 1 - 2 = m - 2 + 1 := by
rw [add_comm, add_comm _ 1, Nat.add_sub_assoc ?_ 1]
omega
rw [hm₂, Finset.sum_range_succ, sum_range_succ]
have hm₃: m - 2 + 1 = m - 1 := by exact id (Eq.symm hm₂)
have hm₄: m - 2 + 2 = m := by exact Eq.symm ((fun {m n} => pred_eq_succ_iff.mp) hm₂)
have hm₅: m - 2 + 3 = m + 1 := by omega
have hm₆: m + 1 - 1 = m := by exact rfl
rw [hm₃, hm₄, hm₅, hm₆]
clear hm₃ hm₄ hm₅ hm₆
rw [add_sq, add_assoc ((∑ x ∈ Finset.range m, a (x + 1) ^ 2) ^ 2)]
have h₅₀: 2 * a (m - 1) ^ 2 * a (m + 1) ^ 2 + 2 * a m ^ 2 * a (m + 1) ^ 2
+ 2 * a (m + 1) ^ 2 * a 1 ^ 2 + 2 * a (m + 1) ^ 2 * a 2 ^ 2 + a (m + 1) ^ 4 ≤
(2 * ∑ x ∈ Finset.range m, a (x + 1) ^ 2) * a (m + 1) ^ 2 + (a (m + 1) ^ 2) ^ 2 := by
rw [← pow_mul]
simp
have h₅₁: 2 * a (m - 1) ^ 2 * a (m + 1) ^ 2 + 2 * a m ^ 2 * a (m + 1) ^ 2 + 2 * a (m + 1) ^ 2 * a 1 ^ 2 +
2 * a (m + 1) ^ 2 * a 2 ^ 2 =
2 * a (m + 1) ^ 2 * (a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2) := by
ring_nf
rw [h₅₁, mul_assoc 2 _ (a (m + 1) ^ 2), mul_comm (∑ x ∈ Finset.range m, a (x + 1) ^ 2), ← mul_assoc 2]
have h₅₂: a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2 ≤ ∑ x ∈ Finset.range m, a (x + 1) ^ 2 := by
exact aux1 a m hm₀
refine mul_le_mul ?_ ?_ ?_ ?_
. exact le_of_eq (by rfl)
. exact h₅₂
. exact _root_.zero_le (a (m - 1) ^ 2 + a m ^ 2 + a 1 ^ 2 + a 2 ^ 2)
. exact _root_.zero_le (2 * a (m + 1) ^ 2)
have h₅₃: ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) +
a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2
≤ (∑ x ∈ Finset.range m, a (x + 1) ^ 2) ^ 2 := by
have h₅₄: ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) +
a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2
≤ ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) +
(a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + 2 * a (m - 1) ^ 2 * a 1 ^ 2) +
(a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2 + 2 * a m ^ 2 * a 2 ^ 2) := by
repeat rw [add_assoc]
repeat refine add_le_add_left ?_ _
have h₅₅: 2 * a (m - 1) ^ 2 * a 1 ^ 2 + (a m ^ 4 + (2 * a m ^ 2 * a 1 ^ 2 + 2 * a m ^ 2 * a 2 ^ 2)) =
(a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2) + (2 * a (m - 1) ^ 2 * a 1 ^ 2 + 2 * a m ^ 2 * a 2 ^ 2) := by
ring_nf
rw [h₅₅]
exact le_self_add
exact le_trans h₅₄ hm₁
apply add_le_add h₅₃ at h₅₀
have h₅₆: ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2)
+ a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + a m ^ 4 + 2 * a m ^ 2 * a 1 ^ 2
+ (2 * a (m - 1) ^ 2 * a (m + 1) ^ 2 + 2 * a m ^ 2 * a (m + 1) ^ 2 + 2 * a (m + 1) ^ 2 * a 1 ^ 2
+ 2 * a (m + 1) ^ 2 * a 2 ^ 2 + a (m + 1) ^ 4)
= ∑ x ∈ Finset.range (m - 2), (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) +
(a (m - 1) ^ 4 + 2 * a (m - 1) ^ 2 * a m ^ 2 + 2 * a (m - 1) ^ 2 * a (m + 1) ^ 2) +
(a m ^ 4 + 2 * a m ^ 2 * a (m + 1) ^ 2 + 2 * a m ^ 2 * a 1 ^ 2) +
(a (m + 1) ^ 4 + 2 * a (m + 1) ^ 2 * a 1 ^ 2 + 2 * a (m + 1) ^ 2 * a 2 ^ 2) := by
repeat rw [add_assoc]
simp
ring_nf
rw [← h₅₆]
exact h₅₀
theorem imo_2007_p6
(a : ℕ → NNReal)
(h₀ : ∑ x ∈ Finset.range 100, ((a (x + 1)) ^ 2) = 1)
(h₁ : ∀ x y, x % 100 = y % 100 → a (x + 1) = a (y + 1)) :
∑ x ∈ Finset.range (99), ((a (x + 1)) ^ 2 * a (x + 2)) + (a 100) ^ 2 * a 1 < (12:NNReal) / (25:NNReal) := by
have h₂: ∀ x, 2 * a x ^ 2 * a (x + 1) * a (x + 2) ≤
(a x * a (x + 1)) ^ 2 + (a x * a (x + 2)) ^ 2 := by
intro x
have h₂₀: 2 * (a x * a (x + 1)) * (a x * a (x + 2)) ≤
(a x * a (x + 1)) ^ 2 + (a x * a (x + 2)) ^ 2 := by
exact two_mul_le_add_sq (a x * a (x + 1)) (a x * a (x + 2))
have h₂₁: 2 * (a x * a (x + 1)) * (a x * a (x + 2)) = 2 * a x ^ 2 * a (x + 1) * a (x + 2) := by
rw [pow_two]
ring_nf
exact le_of_eq_of_le (id (Eq.symm h₂₁)) h₂₀
have h₃: ∀ x ∈ Finset.range 100, a (x + 1) ≤ 1 := by
intros x hx₀
by_contra hx₁
push_neg at hx₁
let fsx : Finset ℕ := {x}
have hx₂: 1 < ∑ x ∈ range 100, a (x + 1) ^ 2 := by
have hx₃: 0 ≤ ∑ x ∈ (range 100 \ fsx), a (x + 1) ^ 2 := by
exact _root_.zero_le (∑ x ∈ range 100 \ fsx, a (x + 1) ^ 2)
have hx₄: 1 < ∑ x ∈ (fsx), a (x + 1) ^ 2 := by
rw [Finset.sum_singleton]
refine one_lt_pow₀ hx₁ ?_
norm_num
have hx₅: ∑ x ∈ (range 100 \ fsx), a (x + 1) ^ 2 + ∑ x ∈ (fsx), a (x + 1) ^ 2 =
∑ x ∈ range 100, a (x + 1) ^ 2 := by
rw [← Finset.sum_union ?_]
. rw [Finset.sdiff_union_self_eq_union]
have hx₆: range 100 ∪ fsx = range 100 := by
refine Finset.union_eq_left.mpr ?_
exact singleton_subset_iff.mpr hx₀
rw [hx₆]
. exact sdiff_disjoint
rw [← hx₅]
exact lt_add_of_nonneg_of_lt hx₃ hx₄
simp_all only [mem_range, lt_self_iff_false]
have h₄: (∑ x ∈ Finset.range 100, (a (x + 2) * (a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3)))) ^ 2 ≤
∑ x ∈ Finset.range 100, (a (x + 1) ^ 4 + 6 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) := by
have h₄₀: (∑ x ∈ Finset.range 100, (a (x + 2) * (a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3)))) ^ 2 ≤
(∑ x ∈ Finset.range 100, (a (x + 2) ^ 2)) *
(∑ x ∈ Finset.range 100, ((a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3))) ^ 2) := by
refine sum_mul_sq_le_sq_mul_sq (range 100) (fun i => a (i + 2)) _
have h₄₁: ∑ x ∈ Finset.range 100, (a (x + 2) ^ 2) = 1 := by
rw [Finset.sum_range_succ'] at h₀
simp at h₀
rw [Finset.sum_range_succ]
have h₄₁₁: a 1 = a 101 := by exact h₁ 0 100 rfl
rw [← h₄₁₁]
exact h₀
have h₄₂: ∑ x ∈ Finset.range 100, ((a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3))) ^ 2 =
∑ x ∈ Finset.range 100, ((a (x + 1) ^ 4 + 4 * a (x + 1) ^ 2 * a (x + 2) * a (x + 3)
+ 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)) := by
refine Finset.sum_congr (rfl) ?_
intros x _
rw [add_sq]
ring_nf
rw [h₄₁, one_mul, h₄₂] at h₄₀
have h₄₃: ∑ x ∈ Finset.range 100, ((a (x + 1) ^ 4 + 4 * a (x + 1) ^ 2 * a (x + 2) * a (x + 3)
+ 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)) ≤
∑ x ∈ Finset.range 100, ((a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * (a (x + 2) ^ 2 + a (x + 3) ^ 2)
+ 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)) := by
refine Finset.sum_le_sum ?_
intros x _
rw [add_comm (a (x + 1) ^ 4) _, add_comm (a (x + 1) ^ 4) _]
rw [add_assoc, add_assoc]
refine add_le_add ?_ ?_
. have hx₁: 2 * a (x + 1) ^ 2 * a (x + 1 + 1) * a (x + 1 + 2) ≤
(a (x + 1) * a (x + 1 + 1)) ^ 2 + (a (x + 1) * a (x + 1 + 2)) ^ 2 := by
exact h₂ (x + 1)
have hx₂: 2 * a (x + 1) ^ 2 * a (x + 2) * a (x + 3) ≤
a (x + 1) ^ 2 * (a (x + 2) ^ 2 + a (x + 3) ^ 2) := by
rw [mul_add]
refine le_of_le_of_eq hx₁ ?_
ring_nf
have hx₃: (4:NNReal) = 2 * 2 := by norm_num
rw [hx₃]
repeat rw [mul_assoc]
have hx₄: 0 < (2:NNReal) := by norm_num
refine (mul_le_mul_left hx₄).mpr ?_
ring_nf
ring_nf at hx₂
exact hx₂
. exact Preorder.le_refl (a (x + 1) ^ 4 + 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)
have h₄₄: ∑ x ∈ Finset.range 100, ((a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * (a (x + 2) ^ 2 + a (x + 3) ^ 2)
+ 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2)) =
∑ x ∈ Finset.range 100, (a (x + 1) ^ 4 + 6 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2
* a (x + 1) ^ 2 * a (x + 3) ^ 2) := by
rw [Finset.sum_add_distrib]
have h₄₄₁: ∑ x ∈ range 100, 4 * a (x + 2) ^ 2 * a (x + 3) ^ 2 =
∑ x ∈ range 100, 4 * a (x + 1) ^ 2 * a (x + 2) ^ 2 := by
rw [Finset.sum_range_succ _ 99, sum_range_succ' _ 99]
have g₀: a 101 = a 1 := by exact h₁ 100 0 rfl
have g₁: a 102 = a 2 := by exact h₁ 101 1 rfl
rw [g₀, g₁]
rw [h₄₄₁, ← Finset.sum_add_distrib]
refine Finset.sum_congr (rfl) ?_
intros x _
rw [mul_add]
ring_nf
rw [h₄₄] at h₄₃
exact le_trans h₄₀ h₄₃
have h₆: ∑ x ∈ range 100, 4 * a (x + 1) ^ 2 * a (x + 2) ^ 2 ≤ 1 := by
have h₆₀: ∑ x ∈ range 100, 4 * a (x + 1) ^ 2 * a (x + 2) ^ 2 =
∑ x ∈ range 100, 4 * (a (x + 1) ^ 2 * a (x + 2) ^ 2) := by
refine Finset.sum_congr rfl ?_
intros x _
ring_nf
rw [h₆₀, ← Finset.mul_sum]
let fs₂ := Finset.range (100)
let fs₀ : Finset ℕ := fs₂.filter (fun x => Odd x)
let fs₁ : Finset ℕ := fs₂.filter (fun x => Even x)
have h₆₁ : Disjoint fs₀ fs₁ := by
refine Finset.sdiff_eq_self_iff_disjoint.mp (by rfl)
have h₆₂ : fs₀ ∪ fs₁ = fs₂ := by
symm
refine Finset.ext_iff.mpr ?_
intro a
constructor
. intro ha₀
refine mem_union.mpr ?mp.a
have ha₁: Odd a ∨ Even a := by exact Or.symm (even_or_odd a)
cases' ha₁ with ha₂ ha₃
. left
refine mem_filter.mpr ?mp.a.inl.h.a
exact And.symm ⟨ha₂, ha₀⟩
. right
refine mem_filter.mpr ?mp.a.inl.h.b
exact And.symm ⟨ha₃, ha₀⟩
. intro ha₀
apply mem_union.mp at ha₀
cases' ha₀ with ha₁ ha₂
. exact mem_of_mem_filter a ha₁
. exact mem_of_mem_filter a ha₂
have h₆₃: 4 * ∑ i ∈ fs₂, a (i + 1) ^ 2 * a (i + 2) ^ 2 ≤
4 * ((∑ i ∈ fs₀, (a (i + 1) ^ 2)) * (∑ i ∈ fs₁, (a (i + 1) ^ 2))) := by
refine mul_le_mul (by norm_num) ?_ ?_ (by norm_num)
. rw [← h₆₂, Finset.sum_union h₆₁]
have g₀: ∑ i ∈ fs₁, a (i + 1) ^ 2 = ∑ i ∈ fs₀, (a i) ^ 2 := by
refine sum_bij ?_ ?h.b2 ?h.b3 ?h.b4 ?h.b5
. intros b _
exact (b + 1)
. intros b hb₀
apply mem_filter.mp at hb₀
cases' hb₀ with hb₀ hb₁
have hb₂: Odd (b + 1) := by exact Even.add_one hb₁
have hb₃: b ≤ 98 := by
by_contra hc₀
apply mem_range.mp at hb₀
interval_cases b
have hc₁: ¬ Even 99 := by decide
exact hc₁ hb₁
have hb₄: b + 1 < 100 := by linarith
have hb₅: (b + 1) ∈ fs₂ := by exact mem_range.mpr hb₄
refine mem_filter.mpr ?_
exact And.symm ⟨hb₂, hb₅⟩
. intros b _ c _ hb₂
linarith
. intros b hb₀
use (b - 1)
refine exists_prop.mpr ?h.a
have hb₁: b ∈ fs₂ ∧ Odd b := by exact mem_filter.mp hb₀
have hb₂: 1 ≤ b := by
by_contra hc
interval_cases b
have hb₃: ¬ Odd 0 := by decide
exact hb₃ hb₁.2
constructor
. cases' hb₁ with hb₁ hb₃
have hb₄: Even (b - 1) := by exact Nat.Odd.sub_odd hb₃ (by decide)
have hb₅: (b - 1) ∈ fs₂ := by
refine mem_range.mpr ?_
have hb₆: b < 100 := by exact List.mem_range.mp hb₁
omega
refine mem_filter.mpr ?_
exact And.symm ⟨hb₄, hb₅⟩
. exact Nat.sub_add_cancel hb₂
. exact fun a_1 _ => rfl
have g₁: ∑ x ∈ fs₁, a (x + 1) ^ 2 * a (x + 2) ^ 2 =
∑ x ∈ fs₀, a (x) ^ 2 * a (x + 1) ^ 2 := by
refine sum_bij ?_ ?_ ?_ ?_ ?_
. intros b _
exact (b + 1)
. intros b hb₀
apply mem_filter.mp at hb₀
cases' hb₀ with hb₀ hb₁
have hb₂: Odd (b + 1) := by exact Even.add_one hb₁
have hb₃: b ≤ 98 := by
by_contra hc₀
apply mem_range.mp at hb₀
interval_cases b
have hc₁: ¬ Even 99 := by decide
exact hc₁ hb₁
have hb₄: b + 1 < 100 := by linarith
have hb₅: (b + 1) ∈ fs₂ := by exact mem_range.mpr hb₄
refine mem_filter.mpr ?_
exact And.symm ⟨hb₂, hb₅⟩
. intros b _ c _ hb₂
linarith
. intros b hb₀
use (b - 1)
refine exists_prop.mpr ?h.b
have hb₁: b ∈ fs₂ ∧ Odd b := by exact mem_filter.mp hb₀
have hb₂: 1 ≤ b := by
by_contra hc
interval_cases b
have hb₃: ¬ Odd 0 := by decide
exact hb₃ hb₁.2
constructor
. cases' hb₁ with hb₁ hb₃
have hb₄: Even (b - 1) := by exact Nat.Odd.sub_odd hb₃ (by decide)
have hb₅: (b - 1) ∈ fs₂ := by
refine mem_range.mpr ?_
have hb₆: b < 100 := by exact List.mem_range.mp hb₁
omega
refine mem_filter.mpr ?_
exact And.symm ⟨hb₄, hb₅⟩
. exact Nat.sub_add_cancel hb₂
. exact fun a_1 _ => rfl
rw [g₀, g₁, Finset.sum_mul_sum, add_comm, ← sum_add_distrib]
refine sum_le_sum ?_
intros x hx₀
apply mem_filter.mp at hx₀
cases' hx₀ with hx₀ hx₁
apply mem_range.mp at hx₀
by_cases hx₃: x < 99
. clear h₀ h₁ h₂ h₃ h₄ h₆₀ g₀ g₁
let fs₃ : Finset ℕ := {x, (x + 2)}
have hx₄: fs₃ ⊆ fs₀ := by
intros b hb₀
have hb₁: b = x ∨ b = x + 2 := by
have g₀: fs₃ = {x, x + 2} := by rfl
simp_all only [mem_insert, mem_singleton]
cases' hb₁ with hb₁ hb₁
. rw [hb₁]
refine mem_filter.mpr ?_
apply mem_range.mpr at hx₀
exact And.symm ⟨hx₁, hx₀⟩
. rw [hb₁]
refine mem_filter.mpr ?_
constructor
. have hx₄: x < 98 := by
by_contra hc
interval_cases x
have hx₅: ¬ Odd 98 := by decide
apply hx₅ hx₁
refine mem_range.mpr ?_
linarith
. refine Odd.add_even hx₁ ?_
decide
have hx₅: ∑ j ∈ fs₃, a (x + 1) ^ 2 * a j ^ 2 = a (x + 1) ^ 2 * a x ^ 2 + a (x + 1) ^ 2 * a (x + 2) ^ 2 := by
have hx₆: fs₃ = {x, x + 2} := by rfl
refine Finset.sum_eq_add_of_mem (x) (x + 2) ?_ ?_ (by norm_num) ?_
. rw [hx₆]
exact mem_insert_self x {x + 2}
. rw [hx₆]
simp
. intros c hc₀ hc₁
exfalso
rw [hx₆] at hc₀
simp only [mem_insert, mem_singleton] at hc₀
have hc₃: ¬ (c ≠ x ∧ c ≠ x + 2) := by
omega
exact hc₃ hc₁
rw [← Finset.sum_sdiff hx₄, hx₅]
refine le_add_left ?_
refine le_of_eq ?_
rw [mul_comm (a x ^ 2) (a (x + 1) ^ 2)]
. interval_cases x
norm_num
have hx₄: a 101 = a 1 := by exact h₁ 100 0 rfl
let fs₃: Finset ℕ := {1, 99}
have hx₅: fs₃ ⊆ fs₀ := by
refine Finset.subset_iff.mpr ?_
intros b hb₀
have hb₁: b = 1 ∨ b = 99 := by exact List.mem_pair.mp hb₀
cases' hb₁ with hb₂ hb₂
. refine mem_filter.mpr ?_
rw [hb₂]
constructor
. refine mem_range.mpr (by decide)
. decide
. rw [hb₂]
refine mem_filter.mpr ?_
constructor
. exact self_mem_range_succ 99
. decide
have hx₆: ∑ x ∈ fs₃, a 100 ^ 2 * a x ^ 2 = a 100 ^ 2 * a 99 ^ 2 + a 100 ^ 2 * a 1 ^ 2 := by
clear h₀ h₁ h₂ h₃ h₄ h₆₀
have hx₇: fs₃ = {1, 99} := by rfl
refine Finset.sum_eq_add_of_mem (99:ℕ) (1:ℕ) ?_ ?_ (by norm_num) ?_
. rw [hx₇]
decide
. rw [hx₇]
decide
. intros c hc₀ hc₁
exfalso
have hc₂: c = 99 ∨ c = 1 := by
refine Or.symm ?_
exact List.mem_pair.mp hc₀
have hc₃: ¬ (c ≠ 99 ∧ c ≠ 1) := by omega
exact hc₃ hc₁
rw [← Finset.sum_sdiff hx₅, hx₄, hx₆]
refine le_add_left ?_
refine le_of_eq ?_
rw [mul_comm (a 99 ^ 2) (a 100 ^ 2)]
. exact _root_.zero_le (∑ i ∈ range 100, a (i + 1) ^ 2 * a (i + 2) ^ 2)
have h₆₄: 4 * ((∑ i ∈ fs₀, (a (i + 1) ^ 2)) * (∑ i ∈ fs₁, (a (i + 1) ^ 2))) ≤
(∑ i ∈ fs₀, (a (i + 1) ^ 2) + ∑ i ∈ fs₁, (a (i + 1) ^ 2)) ^ 2 := by
have g₀: ∀ x y : ℝ, 4 * x * y ≤ (x + y) ^ 2 := by
intros x y
rw [add_sq]
have g₁: 2 * x * y ≤ x ^ 2 + y ^ 2 := by exact two_mul_le_add_sq x y
linarith
rw [← mul_assoc]
let x := (∑ i ∈ fs₀, a (i + 1) ^ 2)
let y := (∑ i ∈ fs₁, a (i + 1) ^ 2)
refine g₀ x y
have h₆₅: (∑ i ∈ fs₀, (a (i + 1) ^ 2) + ∑ i ∈ fs₁, (a (i + 1) ^ 2)) ^ 2 = 1 := by
rw [← Finset.sum_union h₆₁, h₆₂, h₀]
exact one_pow 2
refine le_trans h₆₃ ?_
refine le_trans h₆₄ ?_
rw [h₆₅]
let S : NNReal := ∑ x ∈ Finset.range 99, ((a (x + 1)) ^ 2 * a (x + 2)) + (a 100) ^ 2 * a 1
have hS : S = ∑ x ∈ Finset.range 99, ((a (x + 1)) ^ 2 * a (x + 2)) + (a 100) ^ 2 * a 1 := by rfl
rw [← hS]
have hS₁ : S = ∑ x ∈ Finset.range 100, ((a (x + 1)) ^ 2 * a (x + 2)) := by
rw [Finset.sum_range_succ]
norm_num
have g₀: a 101 = a 1 := by exact h₁ 100 0 rfl
rw [g₀]
have h₇: (3 * S) ^ 2 ≤ 2 := by
have h₇₀: 3 * S = ∑ x ∈ Finset.range 100, (a (x + 2) * (a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3))) := by
have g₀: ∑ x ∈ Finset.range 100, (a (x + 2) * (a (x + 1) ^ 2 + 2 * a (x + 2) * a (x + 3))) =
∑ x ∈ Finset.range 100, (a (x + 1) ^ 2 * a (x + 2) + 2 * a (x + 2) ^ 2 * a (x + 3)) := by
refine Finset.sum_congr rfl ?_
intros x _
ring_nf
have g₁: (3:NNReal) = 1 + 2 := by norm_num
rw [g₀, Finset.sum_add_distrib]
rw [g₁, hS₁, add_mul, one_mul, Finset.mul_sum]
simp
rw [Finset.sum_range_succ' _ 99, sum_range_succ _ 99]
norm_num
have g₂: a 101 = a 1 := by exact h₁ 100 0 rfl
have g₃: a 102 = a 2 := by exact h₁ 101 1 rfl
rw [g₂, g₃, ← mul_assoc 2]
simp
refine Finset.sum_congr rfl ?_
intros x _
ring_nf
rw [← h₇₀] at h₄
refine le_trans h₄ ?_
have h₇₁: ∑ x ∈ range 100, (a (x + 1) ^ 4 + 6 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) =
∑ x ∈ range 100, (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) +
∑ x ∈ range 100, 4 * a (x + 1) ^ 2 * a (x + 2) ^ 2 := by
rw [← Finset.sum_add_distrib]
refine Finset.sum_congr rfl ?_
intros x _
ring_nf
have h₇₂: ∑ x ∈ range 100, (a (x + 1) ^ 4 + 2 * a (x + 1) ^ 2 * a (x + 2) ^ 2 + 2 * a (x + 1) ^ 2 * a (x + 3) ^ 2) ≤ 1 := by
refine le_trans (aux2 a 100 ?_ h₁) ?_
. omega
. refine (sq_le_one_iff₀ ?_).mpr ?_
. exact _root_.zero_le (∑ x ∈ range 100, a (x + 1) ^ 2)
. rw [← h₀]
rw [h₇₁, ← one_add_one_eq_two]
refine add_le_add ?_ h₆
norm_num
exact h₇₂
have h₈ : S ≤ (NNReal.sqrt 2) / (3:NNReal) := by
have h₆₀: NNReal.sqrt (((3:NNReal) * S) ^ 2) ≤ NNReal.sqrt 2 := by
exact NNReal.sqrt_le_sqrt.mpr h₇
rw [sqrt_sq, mul_comm] at h₆₀
refine (le_div_iff₀ (by norm_num)).mpr h₆₀
have h₉: (NNReal.sqrt 2) / (3:NNReal) < (12:NNReal) / (25:NNReal) := by
have h₇₁: 2 < 144 / (625:NNReal) * 9 := by
refine (one_lt_div (by norm_num)).mp ?_
rw [mul_comm_div, ← mul_div_assoc, div_div]
norm_num
refine (one_lt_div (by norm_num)).mpr ?_
norm_num
have h₇₂: (NNReal.sqrt 2 / 3:NNReal) ^ 2 < (12 / 25:NNReal) ^ 2 := by
rw [div_pow, div_pow]
norm_num
refine (div_lt_iff₀ ?_).mpr h₇₁
exact ofNat_pos'
have h₇₃: NNReal.sqrt ((NNReal.sqrt 2 / 3) ^ 2) < NNReal.sqrt ((12 / 25) ^ 2) := by
exact sqrt_lt_sqrt.mpr h₇₂
rw [sqrt_sq, sqrt_sq] at h₇₃
exact h₇₃
exact lt_of_le_of_lt h₈ h₉
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