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namespace OAI
namespace C0Absorption
open scoped BigOperators NNReal ENNReal
noncomputable section
open Finset
section Signs
def realSign (b : Bool) : ℝ := if b then 1 else -1
@[simp] theorem realSign_true : realSign true = 1 := rfl
@[simp] theorem realSign_false : realSign false = -1 := rfl
@[simp] theorem abs_realSign (b : Bool) : |realSign b| = 1 := by cases b <;> norm_num
theorem sum_sign_sq (n : ℕ) (a : Fin n → ℝ) :
∑ ε : Fin n → Bool, (∑ i, realSign (ε i) * a i) ^ 2 =
(2 : ℝ) ^ n * ∑ i, a i ^ 2 := by
induction n with
| zero => simp
| succ n ih =>
rw [← Fintype.sum_equiv (Fin.consEquiv (fun _ : Fin (n + 1) => Bool))
(fun p : Bool × (Fin n → Bool) =>
(∑ i : Fin (n + 1), realSign ((Fin.consEquiv (fun _ => Bool)) p i) * a i) ^ 2)
(fun ε : Fin (n + 1) → Bool => (∑ i, realSign (ε i) * a i) ^ 2)
(fun _ => rfl)]
simp only [Fintype.sum_prod_type, Fin.sum_univ_succ, Fin.consEquiv_apply,
Fin.cons_zero, Fin.cons_succ, Fintype.sum_bool, realSign_true, realSign_false,
one_mul, neg_one_mul]
rw [← Finset.sum_add_distrib]
have heq (x y : ℝ) : (x + y) ^ 2 + (-x + y) ^ 2 = 2 * x ^ 2 + 2 * y ^ 2 := by ring
simp_rw [heq]
rw [Finset.sum_add_distrib, Finset.sum_const, ← Finset.mul_sum, ih]
simp only [Finset.card_univ, Fintype.card_fun, Fintype.card_bool, Fintype.card_fin, nsmul_eq_mul,
Nat.cast_pow, Nat.cast_ofNat, pow_succ]
ring
theorem finite_detector_bound {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X]
(n : ℕ) (m : Fin n → X) (u : Fin n → X →ₗ[ℝ] ℝ)
(α M δ : ℝ) (hα : 0 ≤ α) (_hM : 0 ≤ M)
(hsum : ∀ ε : Fin n → Bool, ‖∑ i, realSign (ε i) • m i‖ ≤ M)
(hdual : ∀ x, α * ∑ k, (u k x) ^ 2 ≤ ‖x‖ ^ 2)
(hdetect : ∀ i, δ ^ 2 ≤ (u i (m i)) ^ 2) :
α * (n : ℝ) * δ ^ 2 ≤ M ^ 2 := by
classical
have hdiag : (n : ℝ) * δ ^ 2 ≤ ∑ k : Fin n, ∑ i : Fin n, (u k (m i)) ^ 2 := by
calc
(n : ℝ) * δ ^ 2 = ∑ k : Fin n, δ ^ 2 := by simp
_ ≤ ∑ k : Fin n, ∑ i : Fin n, (u k (m i)) ^ 2 := by
apply Finset.sum_le_sum
intro k hk
exact (hdetect k).trans (Finset.single_le_sum (fun i _ => sq_nonneg (u k (m i))) (mem_univ k))
have havg : (2 : ℝ) ^ n * (α * ∑ k : Fin n, ∑ i : Fin n, (u k (m i)) ^ 2) ≤
(2 : ℝ) ^ n * M ^ 2 := by
calc
(2 : ℝ) ^ n * (α * ∑ k : Fin n, ∑ i : Fin n, (u k (m i)) ^ 2) =
∑ ε : Fin n → Bool, α * ∑ k : Fin n,
(u k (∑ i : Fin n, realSign (ε i) • m i)) ^ 2 := by
simp only [map_sum, map_smul, smul_eq_mul]
conv_rhs => rw [← Finset.mul_sum, Finset.sum_comm]
simp_rw [sum_sign_sq]
rw [← Finset.mul_sum]
ring
_ ≤ ∑ ε : Fin n → Bool, M ^ 2 := by
apply Finset.sum_le_sum
intro ε hε
exact (hdual _).trans (pow_le_pow_left₀ (norm_nonneg _) (hsum ε) 2)
_ = (2 : ℝ) ^ n * M ^ 2 := by
simp
have ht : α * (∑ k : Fin n, ∑ i : Fin n, (u k (m i)) ^ 2) ≤ M ^ 2 :=
le_of_mul_le_mul_left havg (by positivity)
calc
α * (n : ℝ) * δ ^ 2 = α * ((n : ℝ) * δ ^ 2) := by ring
_ ≤ α * (∑ k : Fin n, ∑ i : Fin n, (u k (m i)) ^ 2) :=
mul_le_mul_of_nonneg_left hdiag hα
_ ≤ M ^ 2 := ht
theorem no_infinite_detectors {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X]
(m : ℕ → X) (u : ℕ → X →ₗ[ℝ] ℝ) (α M δ : ℝ)
(hα : 0 < α) (hM : 0 ≤ M) (hδ : 0 < δ)
(hsum : ∀ n (ε : Fin n → Bool), ‖∑ i, realSign (ε i) • m i‖ ≤ M)
(hdual : ∀ n x, α * ∑ k : Fin n, (u k x) ^ 2 ≤ ‖x‖ ^ 2)
(hdetect : ∀ i, δ ≤ |u i (m i)|) : False := by
have hbound (n : ℕ) : α * (n : ℝ) * δ ^ 2 ≤ M ^ 2 := by
apply finite_detector_bound n (fun i => m i) (fun i => u i) α M δ hα.le hM
(hsum n) (hdual n)
intro i
simpa only [sq_abs] using pow_le_pow_left₀ hδ.le (hdetect i) 2
obtain ⟨n, hn⟩ := exists_nat_gt (M ^ 2 / (α * δ ^ 2))
have hpos : 0 < α * δ ^ 2 := mul_pos hα (sq_pos_of_pos hδ)
have hn' : M ^ 2 < (n : ℝ) * (α * δ ^ 2) := (div_lt_iff₀ hpos).mp hn
nlinarith [hbound n]
def flipSign {n : ℕ} (i : Fin n) : (Fin n → Bool) ≃ (Fin n → Bool) where
toFun ε j := if j = i then !(ε j) else ε j
invFun ε j := if j = i then !(ε j) else ε j
left_inv ε := by funext j; by_cases h : j = i <;> simp [h]
right_inv ε := by funext j; by_cases h : j = i <;> simp [h]
@[simp] theorem realSign_not (b : Bool) : realSign (!b) = -realSign b := by
cases b <;> norm_num
theorem signed_sum_sub_flip {X : Type*} [AddCommGroup X] [Module ℝ X]
{n : ℕ} (m : Fin n → X) (ε : Fin n → Bool) (i : Fin n) :
(∑ j, realSign (ε j) • m j) - (∑ j, realSign (flipSign i ε j) • m j) =
(2 * realSign (ε i)) • m i := by
classical
rw [← Finset.sum_sub_distrib]
rw [Finset.sum_eq_single i]
· simp [flipSign, two_mul, add_smul]
· intro j hj hji
simp [flipSign, hji]
· simp
theorem seminorm_sign_pair {X : Type*} [AddCommGroup X] [Module ℝ X]
(q : Seminorm ℝ X) {n : ℕ} (m : Fin n → X) (ε : Fin n → Bool) (i : Fin n) :
2 * (q (m i)) ^ 2 ≤
(q (∑ j, realSign (ε j) • m j)) ^ 2 +
(q (∑ j, realSign (flipSign i ε j) • m j)) ^ 2 := by
have heq : q ((∑ j, realSign (ε j) • m j) -
(∑ j, realSign (flipSign i ε j) • m j)) = 2 * q (m i) := by
rw [signed_sum_sub_flip, map_smul_eq_mul, Real.norm_eq_abs, abs_mul, abs_realSign]
norm_num
have ht := map_add_le_add q
(∑ j, realSign (ε j) • m j) (-(∑ j, realSign (flipSign i ε j) • m j))
rw [← sub_eq_add_neg, heq, map_neg_eq_map] at ht
have hp := apply_nonneg q (m i)
have ha := apply_nonneg q (∑ j, realSign (ε j) • m j)
have hb := apply_nonneg q (∑ j, realSign (flipSign i ε j) • m j)
nlinarith [sq_nonneg (q (∑ j, realSign (ε j) • m j) -
q (∑ j, realSign (flipSign i ε j) • m j))]
theorem sum_sign_seminorm_sq_ge {X : Type*} [AddCommGroup X] [Module ℝ X]
(q : Seminorm ℝ X) {n : ℕ} (m : Fin n → X) (i : Fin n) :
(2 : ℝ) ^ n * (q (m i)) ^ 2 ≤
∑ ε : Fin n → Bool, (q (∑ j, realSign (ε j) • m j)) ^ 2 := by
classical
have h := Finset.sum_le_sum (s := Finset.univ) (fun ε _ => seminorm_sign_pair q m ε i)
rw [Finset.sum_add_distrib] at h
have heq : (∑ ε : Fin n → Bool, (q (∑ j, realSign (flipSign i ε j) • m j)) ^ 2) =
∑ ε : Fin n → Bool, (q (∑ j, realSign (ε j) • m j)) ^ 2 :=
Fintype.sum_equiv (flipSign i) _ _ (fun _ => rfl)
rw [heq] at h
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fun, Fintype.card_bool,
Fintype.card_fin, nsmul_eq_mul, Nat.cast_pow, Nat.cast_ofNat] at h
linarith
theorem sum_coordinate_choice_le {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X]
(q : ℕ → Seminorm ℝ X)
(hdom : ∀ (s : Finset ℕ) x, ∑ d ∈ s, (q d x) ^ 2 ≤ ‖x‖ ^ 2)
(m : ℕ → X) (M : ℝ)
(hsum : ∀ n (ε : Fin n → Bool), ‖∑ i, realSign (ε i) • m i‖ ≤ M)
(s : Finset ℕ) (index : ℕ → ℕ) :
∑ d ∈ s, (q d (m (index d))) ^ 2 ≤ M ^ 2 := by
classical
let n := s.sup index + 1
have hlt (d : ℕ) (hd : d ∈ s) : index d < n :=
Nat.lt_succ_of_le (Finset.le_sup hd)
have hmain : (2 : ℝ) ^ n * (∑ d ∈ s, (q d (m (index d))) ^ 2) ≤
(2 : ℝ) ^ n * M ^ 2 := by
calc
(2 : ℝ) ^ n * (∑ d ∈ s, (q d (m (index d))) ^ 2) =
∑ d ∈ s, (2 : ℝ) ^ n * (q d (m (index d))) ^ 2 := Finset.mul_sum _ _ _
_ ≤ ∑ d ∈ s, ∑ ε : Fin n → Bool,
(q d (∑ i : Fin n, realSign (ε i) • m i)) ^ 2 := by
apply Finset.sum_le_sum
intro d hd
exact sum_sign_seminorm_sq_ge (q d) (fun i : Fin n => m i) ⟨index d, hlt d hd⟩
_ = ∑ ε : Fin n → Bool, ∑ d ∈ s,
(q d (∑ i : Fin n, realSign (ε i) • m i)) ^ 2 := Finset.sum_comm
_ ≤ ∑ ε : Fin n → Bool, M ^ 2 := by
apply Finset.sum_le_sum
intro ε hε
exact (hdom s _).trans (pow_le_pow_left₀ (norm_nonneg _) (hsum n ε) 2)
_ = (2 : ℝ) ^ n * M ^ 2 := by simp
exact le_of_mul_le_mul_left hmain (by positivity)
theorem sum_ciSup_le_of_forall_choice {ι κ : Type*} [Nonempty κ]
(a : ι → κ → ℝ) (s : Finset ι) (K : ℝ)
(h : ∀ f : ι → κ, ∑ d ∈ s, a d (f d) ≤ K) :
∑ d ∈ s, ⨆ k, a d k ≤ K := by
classical
induction s using Finset.induction_on generalizing K with
| empty => simpa using h (fun _ => Classical.choice (inferInstance : Nonempty κ))
| @insert d s hd ih =>
rw [Finset.sum_insert hd]
rw [← le_sub_iff_add_le]
apply ciSup_le
intro k
have ht : (∑ e ∈ s, ⨆ j, a e j) ≤ K - a d k := by
apply ih
intro f
have hf := h (Function.update f d k)
rw [Finset.sum_insert hd, Function.update_self] at hf
have heq : (∑ e ∈ s, a e (Function.update f d k e)) = ∑ e ∈ s, a e (f e) := by
apply Finset.sum_congr rfl
intro e he
rw [Function.update_of_ne (by intro heq; subst e; exact hd he)]
rw [heq] at hf
linarith
linarith
theorem coordinate_envelope_summable {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X]
(q : ℕ → Seminorm ℝ X)
(hdom : ∀ (s : Finset ℕ) x, ∑ d ∈ s, (q d x) ^ 2 ≤ ‖x‖ ^ 2)
(m : ℕ → X) (M : ℝ)
(hsum : ∀ n (ε : Fin n → Bool), ‖∑ i, realSign (ε i) • m i‖ ≤ M) :
Summable (fun d => ⨆ i : ℕ, (q d (m i)) ^ 2) ∧
(∀ d i, (q d (m i)) ^ 2 ≤ ⨆ j : ℕ, (q d (m j)) ^ 2) := by
have hb (d : ℕ) : BddAbove (Set.range (fun i : ℕ => (q d (m i)) ^ 2)) := by
refine ⟨M ^ 2, ?_⟩
rintro _ ⟨i, rfl⟩
simpa using sum_coordinate_choice_le q hdom m M hsum {d} (fun _ => i)
have hle (d i : ℕ) : (q d (m i)) ^ 2 ≤ ⨆ j : ℕ, (q d (m j)) ^ 2 :=
le_ciSup (hb d) i
constructor
· apply summable_of_sum_le (c := M ^ 2)
· intro d
exact (sq_nonneg _).trans (hle d 0)
· intro s
apply sum_ciSup_le_of_forall_choice (fun d i => (q d (m i)) ^ 2)
exact fun f => sum_coordinate_choice_le q hdom m M hsum s f
· exact hle
theorem square_norm_tendsto_zero {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X]
(q : ℕ → Seminorm ℝ X)
(hq : ∀ x, Summable (fun d => (q d x) ^ 2))
(hnorm : ∀ x, ‖x‖ ^ 2 = ∑' d, (q d x) ^ 2)
(m : ℕ → X) (M : ℝ)
(hsum : ∀ n (ε : Fin n → Bool), ‖∑ i, realSign (ε i) • m i‖ ≤ M)
(hzero : ∀ d, Filter.Tendsto (fun i => q d (m i)) Filter.atTop (nhds 0)) :
Filter.Tendsto (fun i => ‖m i‖ ^ 2) Filter.atTop (nhds 0) := by
have hdom (s : Finset ℕ) (x : X) : ∑ d ∈ s, (q d x) ^ 2 ≤ ‖x‖ ^ 2 := by
rw [hnorm]
exact Summable.sum_le_tsum s (fun _ _ => sq_nonneg _) (hq x)
obtain ⟨hb, hle⟩ := coordinate_envelope_summable q hdom m M hsum
have hp (d : ℕ) : Filter.Tendsto (fun i => (q d (m i)) ^ 2) Filter.atTop (nhds 0) := by
simpa using (hzero d).pow 2
have ht := tendsto_tsum_of_dominated_convergence hb hp
(Filter.Eventually.of_forall fun i d => by
simpa only [Real.norm_of_nonneg (sq_nonneg _)] using hle d i)
simpa only [← hnorm, tsum_zero] using ht
theorem exists_nonzero_coordinate {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X]
(q : ℕ → Seminorm ℝ X)
(hq : ∀ x, Summable (fun d => (q d x) ^ 2))
(hnorm : ∀ x, ‖x‖ ^ 2 = ∑' d, (q d x) ^ 2)
(m : ℕ → X) (M r : ℝ) (hr : 0 < r)
(hlower : ∀ i, r ≤ ‖m i‖)
(hsum : ∀ n (ε : Fin n → Bool), ‖∑ i, realSign (ε i) • m i‖ ≤ M) :
∃ d, ¬ Filter.Tendsto (fun i => q d (m i)) Filter.atTop (nhds 0) := by
by_contra! h
have ht := square_norm_tendsto_zero q hq hnorm m M hsum h
have hl : r ^ 2 ≤ (0 : ℝ) := ge_of_tendsto ht
(Filter.Eventually.of_forall fun i => pow_le_pow_left₀ hr.le (hlower i) 2)
nlinarith [sq_pos_of_pos hr]
end Signs
end
end C0Absorption
end OAI
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